1MA 03. NOMBRES COMPLEXOS/1MA.03.1 Unitat i, EqG2 en ℂ 1MA.03.1.10DT Eq 2n grau ax^2+c Equació «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mrow»«msup mathcolor=¨#FFFFC3¨»«mi mathcolor=¨#FFFFC3¨»ax«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»c«/mi»«mo mathcolor=¨#FFFFC3¨»§#160;«/mo»«mo mathcolor=¨#FFFFC3¨»=«/mo»«mo mathcolor=¨#FFFFC3¨»§#160;«/mo»«mn mathcolor=¨#FFFFC3¨»0«/mn»«mo mathcolor=¨#FFFFC3¨»§#160;«/mo»«mo mathcolor=¨#FFFFC3¨»(«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»a«/mi»«mo mathcolor=¨#FFFFC3¨»§#183;«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»c«/mi»«mo mathcolor=¨#FFFFC3¨»§#62;«/mo»«mn mathcolor=¨#FFFFC3¨»0«/mn»«mo mathcolor=¨#FFFFC3¨»)«/mo»«/mrow»«/mstyle»«/math»en ℂ
Si l'equació és del tipus ax² + c = 0, i a·c>0, aïlla la x:
«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»ax«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»+«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»0«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8660;«/mo»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»ax«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»c«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#8660;«/mo»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mfrac mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»c«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«/mrow»«/mstyle»«/math»

Com que -c/a és negatiu, es transforma en «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mfrac mathcolor=¨#003300¨»«mi mathvariant=¨bold¨»c«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#183;«/mo»«msup mathcolor=¨#FF0000¨»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»i«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mstyle»«/math»
I les solucions són: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#177;«/mo»«msqrt mathcolor=¨#003300¨»«mfrac»«mi mathvariant=¨bold¨»c«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«mo mathvariant=¨bold¨»§#183;«/mo»«msup»«mi mathvariant=¨bold¨»i«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#177;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«msqrt mathcolor=¨#003300¨»«mfrac»«mi mathvariant=¨bold¨»c«/mi»«mi mathvariant=¨bold¨»a«/mi»«/mfrac»«/msqrt»«/mrow»«/mstyle»«/math»

La fracció i l'arrel es simplifiquen, si es pot.

]]> 0.0000000 0.0000000 0 1MA.03.1.11Q Eq 2G ax^2+c (enters) Troba les solucions de l'equació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»11«/mn»«/mrow»«/mstyle»«/math» en el conjunt dels complexos.

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo»-«/mo»«mfrac»«mrow»«mi»i«/mi»«mo»§#183;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«mn»2«/mn»«/mfrac»«mo»,«/mo»«mfrac»«mrow»«mi»i«/mi»«mo»§#183;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§#160;«/mo»«mo»(«/mo»«mi»c«/mi»«mi»l«/mi»«mi»a«/mi»«mi»u«/mi»«mi»s«/mi»«mo»)«/mo»«/mstyle»«/math» racionalitzada i simplificada.

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>a1</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><msup><mi>a1</mi><mn>2</mn></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>s</mi><mo>*</mo><mi>a</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mo>-</mo><mi>s</mi><mo>*</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mi>c_1</mi><mi>a_1</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mo>-</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>=</mo><msqrt><mi>i</mi></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>=</mo><mi>s</mi><mo>*</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>s</mi><mo>*</mo><mi>c</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>s</mi><mo>*</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>s</mi><mo>*</mo><mi>c</mi><mo>=</mo><mn>0</mn><mo>,</mo><complexes/></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>,</mo><mn>16</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>16</mn><mn>25</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>16</mn><mn>25</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>5</mn></mfrac></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>25</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>16</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>*</mo><imaginaryi/></mrow><mn>5</mn></mfrac></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>4</mn><mo>*</mo><imaginaryi/></mrow><mn>5</mn></mfrac></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfrac><mrow><mn>4</mn><mo>*</mo><imaginaryi/></mrow><mn>5</mn></mfrac><mo>,</mo><mo>-</mo><mfrac><mrow><mn>4</mn><mo>*</mo><imaginaryi/></mrow><mn>5</mn></mfrac></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="check_rationalized"/><assertion name="check_simplified"/><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Aïllo x²: #a_1 x² = #c_1 i divideixo per #a_1: x² = #h .
Com que és negatiu, el transformo en: x² = #i i²
que té dues arrels.

]]> 1MA.03.1.15Q Eq 2G ax^2+c (irracional) Troba les solucions de l'equació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»11«/mn»«/mrow»«/mstyle»«/math» en el conjunt dels complexos.

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo»-«/mo»«mfrac»«mrow»«mi»i«/mi»«mo»§#183;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«mn»2«/mn»«/mfrac»«mo»,«/mo»«mfrac»«mrow»«mi»i«/mi»«mo»§#183;«/mo»«msqrt»«mn»6«/mn»«/msqrt»«/mrow»«mn»2«/mn»«/mfrac»«/mrow»«/mfenced»«mo»§#160;«/mo»«mo»(«/mo»«mi»c«/mi»«mi»l«/mi»«mi»a«/mi»«mi»u«/mi»«mi»s«/mi»«mo»)«/mo»«/mstyle»«/math» racionalitzada i simplificada.

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>2</mn><mo>,</mo><mn>6</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup></mtd></mtr></mtable></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>s</mi><mo>=</mo><mi>aleatori</mi><mfenced><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn></mtd></mtr></mtable></mfenced></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a_1</mi><mo>=</mo><mi>s</mi><mo>*</mo><mi>a</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c_1</mi><mo>=</mo><mo>-</mo><mi>s</mi><mo>*</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><mfrac><mi>c_1</mi><mi>a_1</mi></mfrac></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi><mo>=</mo><mo>-</mo><mi>h</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi><mo>=</mo><msqrt><mi>i</mi></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>=</mo><mi>s</mi><mo>*</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>s</mi><mo>*</mo><mi>c</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>s</mi><mo>*</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>s</mi><mo>*</mo><mi>c</mi><mo>=</mo><mn>0</mn><mo>,</mo><complexes/></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>,</mo><mn>4</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>4</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>4</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mo>-</mo><imaginaryi/></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><imaginaryi/></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mo>-</mo><imaginaryi/><mo>,</mo><imaginaryi/></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="check_rationalized"/><assertion name="check_simplified"/><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Aïllo x²: #a_1 x² = #c_1 i divideixo per #a_1: x² = #h .
Com que és negatiu, el transformo en: x² = #i i²
que té dues arrels.

]]> 1MA.03.1.40DT E2G ax2+bx+c (Δ

Equació «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨18px¨»«mrow»«msup mathcolor=¨#FFFFC3¨»«mi mathcolor=¨#FFFFC3¨»ax«/mi»«mn»2«/mn»«/msup»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathcolor=¨#FFFFC3¨»bx«/mi»«mo mathcolor=¨#FFFFC3¨»+«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»c«/mi»«mo mathcolor=¨#FFFFC3¨»§#160;«/mo»«mo mathcolor=¨#FFFFC3¨»(«/mo»«mi mathvariant=¨normal¨ mathcolor=¨#FFFFC3¨»§#916;«/mi»«mo mathcolor=¨#FFFFC3¨»§#60;«/mo»«mn mathcolor=¨#FFFFC3¨»0«/mn»«mo mathcolor=¨#FFFFC3¨»)«/mo»«/mrow»«/mstyle»«/math» en ℂ

En el conjunt dels complexos, les equacions de 2n grau es fan igual que en el conjunt dels reals, però si el discriminant és negatiu (Δ=-K), es substitueix per Δ=Ki².
Es calcula primer el discriminant: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»§#916;«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»§#160;«/mo»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»b«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»-«/mo»«mfenced mathcolor=¨#003300¨ open=¨[¨ close=¨]¨»«mrow»«mn mathvariant=¨bold¨»4«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»a«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»c«/mi»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» = -K
Com que és negatiu, el substitueix per Ki².
Les solucions són «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»z«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»§#177;«/mo»«msqrt»«mi mathvariant=¨bold¨»K«/mi»«msup mathcolor=¨#FF0000¨»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»i«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/msqrt»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»=«/mo»«mfrac mathcolor=¨#003300¨»«mrow»«mo mathvariant=¨bold¨»-«/mo»«mi mathvariant=¨bold¨»b«/mi»«mo mathvariant=¨bold¨»§#177;«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#FF0000¨»i«/mi»«msqrt»«mi mathvariant=¨bold¨»K«/mi»«/msqrt»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«/mstyle»«/math»

]]> 0.0000000 0.0000000 0 1MA.03.1.41Q Eq2G (CoefEnters) Troba les solucions de l'equació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»11«/mn»«/mstyle»«/math» en el conjunt dels complexos.

Format de la resposta: {3-2i,3+3i} amb CLAUS

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>a</mi><mo>*</mo><mo>(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>)</mo><mo><</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>a</mi><mo>*</mo><mo>(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mo>-</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msqrt><mi>g</mi></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><complexes/></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>z</mi><mo>,</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1764</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1764</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>42</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>,</mo><mi>b1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>42</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>294</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo>-</mo><mn>42</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>7</mn><mo>-</mo><mn>7</mn><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>7</mn><mo>+</mo><mn>7</mn><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>-</mo><mn>7</mn><mo>*</mo><imaginaryi/></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>+</mo><mn>7</mn><mo>*</mo><imaginaryi/></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>7</mn><mo>-</mo><mn>7</mn><mo>*</mo><imaginaryi/><mo>,</mo><mn>7</mn><mo>+</mo><mn>7</mn><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Com que el discriminant, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#916;«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«/math»#d, és negatiu el transformem en #g * i², que té dues arrels ± #h * i. Això ens permet calcular les dues solucions amb l'expressió:

«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»x«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mfrac mathcolor=¨#0000FF¨»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«mn mathvariant=¨bold¨»1«/mn»«mo mathvariant=¨bold¨»§#177;«/mo»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»h«/mi»«mo mathvariant=¨bold¨»§#183;«/mo»«mi mathvariant=¨bold¨»i«/mi»«/mrow»«mrow»«mn mathvariant=¨bold¨»2«/mn»«mo mathvariant=¨bold¨»*«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math»

]]> 1MA.03.1.42Q Eq2G (CoefEnters)_2 Troba les solucions de l'equació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»11«/mn»«/mstyle»«/math» en el conjunt dels complexos.

Format de la resposta: {3-2i,3+3i} amb CLAUS

]]> 1.0000000 0.5000000 0 0 #sol <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>9</mn><mo>,</mo><mn>9</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>1</mn><mo>,</mo><mn>9</mn><mo>)</mo><mo>*</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>a</mi><mo>*</mo><mo>(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>)</mo><mo><</mo><mn>0</mn></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><msup><mfenced><mrow><mo>-</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi></mrow></mfenced><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>a</mi><mo>*</mo><mo>(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>+</mo><msup><mi>c</mi><mn>2</mn></msup><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi><mo>=</mo><mo>-</mo><mi>d</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi><mo>=</mo><msqrt><mi>g</mi></msqrt></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>=</mo><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>a</mi><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>+</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>*</mo><mo>(</mo><mi>x</mi><mo>-</mo><mo>(</mo><mi>b</mi><mo>-</mo><mi>c</mi><mo>*</mo><imaginaryi/><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn><mo>,</mo><complexes/></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi><mo>=</mo><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi><mo>=</mo><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>z</mi><mo>,</mo><mi>t</mi></mtd></mtr></mtable></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b1</mi><mo>=</mo><mn>2</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>b</mi></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>,</mo><mn>7</mn><mo>,</mo><mn>7</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1764</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>g</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1764</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>h</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>42</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>,</mo><mi>b1</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>3</mn><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>42</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>294</mn><mo>=</mo><mn>0</mn><mo>,</mo><mo>-</mo><mn>42</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>7</mn><mo>-</mo><mn>7</mn><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced><mo>,</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mi>x</mi><mo>=</mo><mn>7</mn><mo>+</mo><mn>7</mn><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced></mtd></mtr></mtable></mfenced></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>z</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>-</mo><mn>7</mn><mo>*</mo><imaginaryi/></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>t</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>7</mn><mo>+</mo><mn>7</mn><mo>*</mo><imaginaryi/></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>7</mn><mo>-</mo><mn>7</mn><mo>*</mo><imaginaryi/><mo>,</mo><mn>7</mn><mo>+</mo><mn>7</mn><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer>#sol</correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">textField</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Com que el discriminant, «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»§#916;«/mi»«mo»§nbsp;«/mo»«mo»=«/mo»«/math»#d, és negatiu el transformem en #g * i², que té dues arrels ± #h * i. Això ens permet calcular les dues solucions amb l'expressió:

]]> 1MA.03.1.51Q Eq2G (coefQualssevol) Troba les solucions de l'equació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»11«/mn»«/mrow»«/mstyle»«/math» en el conjunt dels complexos.

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«msqrt»«mn»467«/mn»«/msqrt»«mo»§#183;«/mo»«mi»i«/mi»«/mrow»«mn»6«/mn»«/mfrac»«mo»,«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«msqrt»«mn»467«/mn»«/msqrt»«mo»§#183;«/mo»«mi»i«/mi»«/mrow»«mn»6«/mn»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math» amb claus

]]> 1.0000000 0.5000000 0 0 #sol]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="es">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><apply><csymbol definitionURL="http://www.wiris.com/XML/csymbol">repeat</csymbol><mtable><mtr><mtd><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>{</mo><mo>-</mo><mn>3</mn><mo>,</mo><mo>-</mo><mn>2</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>}</mo><mo>)</mo></mtd></mtr><mtr><mtd><mi>b</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>10</mn><mo>,</mo><mn>10</mn><mo>)</mo></mtd></mtr><mtr><mtd><mi>c</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mo>-</mo><mn>40</mn><mo>,</mo><mn>40</mn><mo>)</mo></mtd></mtr><mtr><mtd/></mtr></mtable><mrow><mo>(</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>c</mi><mo><</mo><mn>0</mn><mo>)</mo></mrow></apply></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi><mo>=</mo><msup><mi>b</mi><mn>2</mn></msup><mo>-</mo><mn>4</mn><mo>*</mo><mi>a</mi><mo>*</mo><mi>c</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D11</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mi>D</mi></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi><mo>=</mo><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>=</mo><mi>resol</mi><mfenced><mrow><mi>a</mi><mo>*</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>b</mi><mo>*</mo><mi>x</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn><mo>,</mo><complexes/></mrow></mfenced></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi><mo>=</mo><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><msub><mi>A</mi><mrow><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>A</mi><mrow><mn>2</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></mtd></mtr></mtable></mfenced></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>1</mn><mo>,</mo><mn>6</mn><mo>,</mo><mo>-</mo><mn>17</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e_11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>6</mn><mo>*</mo><mi>x</mi><mo>-</mo><mn>17</mn><mo>=</mo><mn>0</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mn>32</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>D11</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>32</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced close="}" open="{"><mtable align="center"><mtr><mtd><mn>3</mn><mo>-</mo><mn>2</mn><mo>*</mo><msqrt><mn>2</mn></msqrt><mo>*</mo><imaginaryi/><mo>,</mo><mn>3</mn><mo>+</mo><mn>2</mn><mo>*</mo><msqrt><mn>2</mn></msqrt><mo>*</mo><imaginaryi/></mtd></mtr></mtable></mfenced></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mspace linebreak="newline"/></math>]]></correctAnswer></correctAnswers><assertions><assertion name="equivalent_symbolic"/><assertion name="syntax_expression"/></assertions><options><option name="tolerance">10^(-4)</option><option name="relative_tolerance">false</option><option name="precision">8</option><option name="implicit_times_operator">false</option><option name="times_operator">·</option><option name="imaginary_unit">i</option></options><localData><data name="inputField">inlineEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> El discriminant és #D. Com que és negatiu cal emprar el fet que i² = -1 i per tant el discriminant és «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»D«/mi»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#8201;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»D«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#0000FF¨»11«/mn»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#183;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#0000FF¨»§#160;«/mo»«msup mathcolor=¨#0000FF¨»«mi mathvariant=¨bold¨ mathcolor=¨#0000FF¨»i«/mi»«mn mathvariant=¨bold¨»2«/mn»«/msup»«/mrow»«/mstyle»«/math»

]]> 1MA.03.1.52Q Eq2G (coefQualssevol)_2 Troba les solucions de l'equació: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«mo mathvariant=¨bold¨ mathcolor=¨#003300¨»#«/mo»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»e_«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#003300¨»11«/mn»«/mrow»«/mstyle»«/math» en el conjunt dels complexos.

Format de la resposta: «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mfenced open=¨{¨ close=¨}¨»«mrow»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»+«/mo»«mfrac»«mrow»«msqrt»«mn»467«/mn»«/msqrt»«mo»§#183;«/mo»«mi»i«/mi»«/mrow»«mn»6«/mn»«/mfrac»«mo»,«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mo»-«/mo»«mfrac»«mrow»«msqrt»«mn»467«/mn»«/msqrt»«mo»§#183;«/mo»«mi»i«/mi»«/mrow»«mn»6«/mn»«/mfrac»«/mrow»«/mfenced»«/mstyle»«/math» amb claus

]]> 1MA.03.1.60DT POTÈNCIES DE i Potències de i Com que: i = i; i² = -1; i³ = -i; i⁴ = 1

per calcular les potències de i, cal fer la divisió entera de l'exponent per 4 i fixar-se en el residu r:

Exemple 1: i²⁶. 26:4 → 26 = 4·6 +2 (residu 2):

i²⁶ = (i⁴)⁶ · i² = 1 · i² = -1

Exemple 2: i³⁹: 39 = 4·9 + 3 => i³⁹ = (i⁴)⁹ ·i³ = 1·i³ = -i.

]]> 0.0000000 0.0000000 0 1MA.03.1.61Q Potències i Calcula les següents potències de la unitat imaginària, i:

a. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/msup»«/mstyle»«/math»

b. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/msup»«/mstyle»«/math»

c. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msup»«/mstyle»«/math»

^{d. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«/mrow»«/msup»«/mstyle»«/math»}

]]> 1.0000000 0.3333333 0 0 a. =#sol1b. =#sol2c. =#sol3d. =#sol4]]> <question><wirisCasSession><![CDATA[<session lang="ca" version="2.0"><library closed="false"><mtext style="color:#ffc800" xml:lang="ca">variables</mtext><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>=</mo><mi>aleatori</mi><mo>(</mo><mn>200</mn><mo>,</mo><mn>400</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>b</mi><mo>=</mo><mi>a</mi><mo>+</mo><mn>17</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi><mo>=</mo><mi>a</mi><mo>-</mo><mn>98</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mi>a</mi><mo>+</mo><mn>215</mn></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>=</mo><mi>residu</mi><mo>(</mo><mi>a</mi><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r2</mi><mo>=</mo><mi>residu</mi><mo>(</mo><mi>b</mi><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r3</mi><mo>=</mo><mi>residu</mi><mo>(</mo><mi>c</mi><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r4</mi><mo>=</mo><mi>residu</mi><mo>(</mo><mi>d</mi><mo>,</mo><mn>4</mn><mo>)</mo></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>=</mo><msup><imaginaryi/><mi>a</mi></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol2</mi><mo>=</mo><msup><imaginaryi/><mi>b</mi></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol3</mi><mo>=</mo><msup><imaginaryi/><mi>c</mi></msup></math></input></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol4</mi><mo>=</mo><msup><imaginaryi/><mi>d</mi></msup></math></input></command></group></library><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>b</mi><mo>,</mo><mi>c</mi><mo>,</mo><mi>d</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>335</mn><mo>,</mo><mn>352</mn><mo>,</mo><mn>237</mn><mo>,</mo><mn>550</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>sol1</mi><mo>,</mo><mi>sol2</mi><mo>,</mo><mi>sol3</mi><mo>,</mo><mi>sol4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><imaginaryi/><mo>,</mo><mn>1</mn><mo>,</mo><imaginaryi/><mo>,</mo><mo>-</mo><mn>1</mn></math></output></command><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>r1</mi><mo>,</mo><mi>r2</mi><mo>,</mo><mi>r3</mi><mo>,</mo><mi>r4</mi></math></input><output><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></output></command></group><group><command><input><math xmlns="http://www.w3.org/1998/Math/MathML"/></input></command></group></session>]]></wirisCasSession><correctAnswers><correctAnswer type="mathml"><![CDATA[<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>1</mn><mspace linebreak="newline"/><mi>b</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>2</mn><mspace linebreak="newline"/><mi>c</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>3</mn><mspace linebreak="newline"/><mi mathvariant="normal">d</mi><mo>.</mo><mo> </mo><mo>=</mo><mo>#</mo><mi>s</mi><mi>o</mi><mi>l</mi><mn>4</mn></math>]]></correctAnswer></correctAnswers><assertions><assertion name="syntax_expression"/><assertion name="equivalent_symbolic"/></assertions><options><option name="tolerance">10^(-3)</option><option name="relative_tolerance">true</option><option name="precision">4</option><option name="times_operator">·</option><option name="implicit_times_operator">false</option><option name="imaginary_unit">i</option></options><localData><data name="inputCompound">true</data><data name="inputField">popupEditor</data><data name="gradeCompound">and</data><data name="gradeCompoundDistribution"></data><data name="casSession"/></localData></question> Com que les potències de i es repeteixen,

i^4·q+r = i^4·q · i^r = 1 · i^r

Cal doncs fer la divisió entera de la potència per 4.

El residu de la divisió de #a per 4 és #r1, per això «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«mrow»«msup mathcolor=¨#00007F¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»§#160;«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«msup mathcolor=¨#00007F¨»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»r«/mi»«mn mathvariant=¨bold¨»1«/mn»«/mrow»«/msup»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»=«/mo»«mo mathvariant=¨bold¨ mathcolor=¨#00007F¨»#«/mo»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»s«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»o«/mi»«mi mathvariant=¨bold-italic¨ mathcolor=¨#00007F¨»l«/mi»«mn mathvariant=¨bold¨ mathcolor=¨#00007F¨»1«/mn»«/mrow»«/mstyle»«/math»

]]> El residu de la divisió de #b per 4 és #r2

El residu de la divisió de #c per 4 és #r3

El residu de la divisió de #d per 4 és #r4

]]> 1MA.03.1.62Q Potències i_2 Calcula les següents potències de la unitat imaginària, i:

a. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»a«/mi»«/mrow»«/msup»«/mstyle»«/math»

b. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»b«/mi»«/mrow»«/msup»«/mstyle»«/math»

c. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#003300¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»c«/mi»«mo mathvariant=¨bold¨»§#160;«/mo»«/mrow»«/msup»«/mstyle»«/math»

^{d. «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨12px¨»«msup mathcolor=¨#003300¨»«mi mathvariant=¨bold¨ mathcolor=¨#007F00¨»i«/mi»«mrow»«mo mathvariant=¨bold¨»#«/mo»«mi mathvariant=¨bold¨»d«/mi»«/mrow»«/msup»«/mstyle»«/math»}

i^4·q+r = i^4·q · i^r = 1 · i^r

Cal doncs fer la divisió entera de la potència per 4.

]]> El residu de la divisió de #b per 4 és #r2

El residu de la divisió de #c per 4 és #r3

El residu de la divisió de #d per 4 és #r4

]]>