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(4+2*i)*x+(5-3*I)*x^2=13+i la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
                         2         
(4 + 2*I)*x + (5 - 3*I)*x  = 13 + I
x2(5−3i)+x(4+2i)=13+ix^{2} \left(5 - 3 i\right) + x \left(4 + 2 i\right) = 13 + i
Simplificar
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.

La ecuación se convierte de
x2(5−3i)+x(4+2i)=13+ix^{2} \left(5 - 3 i\right) + x \left(4 + 2 i\right) = 13 + i
en
(x2(5−3i)+x(4+2i))+(−13−i)=0\left(x^{2} \left(5 - 3 i\right) + x \left(4 + 2 i\right)\right) + \left(-13 - i\right) = 0
Abramos la expresión en la ecuación
(x2(5−3i)+x(4+2i))+(−13−i)=0\left(x^{2} \left(5 - 3 i\right) + x \left(4 + 2 i\right)\right) + \left(-13 - i\right) = 0
Obtenemos la ecuación cuadrática
5x2−3ix2+4x+2ix−13−i=05 x^{2} - 3 i x^{2} + 4 x + 2 i x - 13 - i = 0
Es la ecuación de la forma
a*x^2 + b*x + c = 0

La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=5−3ia = 5 - 3 i
b=4+2ib = 4 + 2 i
c=−13−ic = -13 - i
, entonces
D = b^2 - 4 * a * c = 

(4 + 2*i)^2 - 4 * (5 - 3*i) * (-13 - i) = (4 + 2*i)^2 - (-13 - i)*(20 - 12*i)

La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

o
x1=(10+6i)(−4+−(−13−i)(20−12i)+(4+2i)2−2i)136x_{1} = \frac{\left(10 + 6 i\right) \left(-4 + \sqrt{- \left(-13 - i\right) \left(20 - 12 i\right) + \left(4 + 2 i\right)^{2}} - 2 i\right)}{136}
x2=(10+6i)(−4−2i−−(−13−i)(20−12i)+(4+2i)2)136x_{2} = \frac{\left(10 + 6 i\right) \left(-4 - 2 i - \sqrt{- \left(-13 - i\right) \left(20 - 12 i\right) + \left(4 + 2 i\right)^{2}}\right)}{136}
Teorema de Cardano-Vieta
reescribamos la ecuación
x2(5−3i)+x(4+2i)=13+ix^{2} \left(5 - 3 i\right) + x \left(4 + 2 i\right) = 13 + i
de
ax2+bx+c=0a x^{2} + b x + c = 0
como ecuación cuadrática reducida
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
(5+3i)(x2(5−3i)+x(4+2i)−13−i)34=0\frac{\left(5 + 3 i\right) \left(x^{2} \left(5 - 3 i\right) + x \left(4 + 2 i\right) - 13 - i\right)}{34} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=(4+2i)(5+3i)34p = \frac{\left(4 + 2 i\right) \left(5 + 3 i\right)}{34}
q=caq = \frac{c}{a}
q=(−13−i)(5+3i)34q = \frac{\left(-13 - i\right) \left(5 + 3 i\right)}{34}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−(4+2i)(5+3i)34x_{1} + x_{2} = - \frac{\left(4 + 2 i\right) \left(5 + 3 i\right)}{34}
x1x2=(−13−i)(5+3i)34x_{1} x_{2} = \frac{\left(-13 - i\right) \left(5 + 3 i\right)}{34}
Gráfica
Respuesta rápida [src] 
              /                     /    /30\\                 /    /30\\\                 /    /30\\                 /    /30\\
              |                     |atan|--||                 |atan|--|||                 |atan|--||                 |atan|--||
              |         4 ______    |    \71/|     4 ______    |    \71/||     4 ______    |    \71/|     4 ______    |    \71/|
              |       5*\/ 5941 *sin|--------|   3*\/ 5941 *cos|--------||   3*\/ 5941 *sin|--------|   5*\/ 5941 *cos|--------|
       7      |  11                 \   2    /                 \   2    /|                 \   2    /                 \   2    /
x1 = - -- + I*|- -- - ------------------------ + ------------------------| + ------------------------ + ------------------------
       34     \  34              34                         34           /              34                         34
x1=−734+359414sin⁡(atan⁡(3071)2)34+559414cos⁡(atan⁡(3071)2)34+i(−1134−559414sin⁡(atan⁡(3071)2)34+359414cos⁡(atan⁡(3071)2)34)x_{1} = - \frac{7}{34} + \frac{3 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{5 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + i \left(- \frac{11}{34} - \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right) 
              /                     /    /30\\                 /    /30\\\                 /    /30\\                 /    /30\\
              |                     |atan|--||                 |atan|--|||                 |atan|--||                 |atan|--||
              |         4 ______    |    \71/|     4 ______    |    \71/||     4 ______    |    \71/|     4 ______    |    \71/|
              |       3*\/ 5941 *cos|--------|   5*\/ 5941 *sin|--------||   5*\/ 5941 *cos|--------|   3*\/ 5941 *sin|--------|
       7      |  11                 \   2    /                 \   2    /|                 \   2    /                 \   2    /
x2 = - -- + I*|- -- - ------------------------ + ------------------------| - ------------------------ - ------------------------
       34     \  34              34                         34           /              34                         34
x2=−559414cos⁡(atan⁡(3071)2)34−734−359414sin⁡(atan⁡(3071)2)34+i(−359414cos⁡(atan⁡(3071)2)34−1134+559414sin⁡(atan⁡(3071)2)34)x_{2} = - \frac{5 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{7}{34} - \frac{3 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + i \left(- \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{11}{34} + \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right) 
x2 = -5*5941^(1/4)*cos(atan(30/71)/2)/34 - 7/34 - 3*5941^(1/4)*sin(atan(30/71)/2)/34 + i*(-3*5941^(1/4)*cos(atan(30/71)/2)/34 - 11/34 + 5*5941^(1/4)*sin(atan(30/71)/2)/34)
Suma y producto de raíces [src] 
suma
         /                     /    /30\\                 /    /30\\\                 /    /30\\                 /    /30\\            /                     /    /30\\                 /    /30\\\                 /    /30\\                 /    /30\\
         |                     |atan|--||                 |atan|--|||                 |atan|--||                 |atan|--||            |                     |atan|--||                 |atan|--|||                 |atan|--||                 |atan|--||
         |         4 ______    |    \71/|     4 ______    |    \71/||     4 ______    |    \71/|     4 ______    |    \71/|            |         4 ______    |    \71/|     4 ______    |    \71/||     4 ______    |    \71/|     4 ______    |    \71/|
         |       5*\/ 5941 *sin|--------|   3*\/ 5941 *cos|--------||   3*\/ 5941 *sin|--------|   5*\/ 5941 *cos|--------|            |       3*\/ 5941 *cos|--------|   5*\/ 5941 *sin|--------||   5*\/ 5941 *cos|--------|   3*\/ 5941 *sin|--------|
  7      |  11                 \   2    /                 \   2    /|                 \   2    /                 \   2    /     7      |  11                 \   2    /                 \   2    /|                 \   2    /                 \   2    /
- -- + I*|- -- - ------------------------ + ------------------------| + ------------------------ + ------------------------ + - -- + I*|- -- - ------------------------ + ------------------------| - ------------------------ - ------------------------
  34     \  34              34                         34           /              34                         34                34     \  34              34                         34           /              34                         34
(−559414cos⁡(atan⁡(3071)2)34−734−359414sin⁡(atan⁡(3071)2)34+i(−359414cos⁡(atan⁡(3071)2)34−1134+559414sin⁡(atan⁡(3071)2)34))+(−734+359414sin⁡(atan⁡(3071)2)34+559414cos⁡(atan⁡(3071)2)34+i(−1134−559414sin⁡(atan⁡(3071)2)34+359414cos⁡(atan⁡(3071)2)34))\left(- \frac{5 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{7}{34} - \frac{3 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + i \left(- \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{11}{34} + \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right)\right) + \left(- \frac{7}{34} + \frac{3 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{5 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + i \left(- \frac{11}{34} - \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right)\right) 
=
         /                     /    /30\\                 /    /30\\\     /                     /    /30\\                 /    /30\\\
         |                     |atan|--||                 |atan|--|||     |                     |atan|--||                 |atan|--|||
         |         4 ______    |    \71/|     4 ______    |    \71/||     |         4 ______    |    \71/|     4 ______    |    \71/||
         |       5*\/ 5941 *sin|--------|   3*\/ 5941 *cos|--------||     |       3*\/ 5941 *cos|--------|   5*\/ 5941 *sin|--------||
  7      |  11                 \   2    /                 \   2    /|     |  11                 \   2    /                 \   2    /|
- -- + I*|- -- - ------------------------ + ------------------------| + I*|- -- - ------------------------ + ------------------------|
  17     \  34              34                         34           /     \  34              34                         34           /
−717+i(−359414cos⁡(atan⁡(3071)2)34−1134+559414sin⁡(atan⁡(3071)2)34)+i(−1134−559414sin⁡(atan⁡(3071)2)34+359414cos⁡(atan⁡(3071)2)34)- \frac{7}{17} + i \left(- \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{11}{34} + \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right) + i \left(- \frac{11}{34} - \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right) 
producto
/         /                     /    /30\\                 /    /30\\\                 /    /30\\                 /    /30\\\ /         /                     /    /30\\                 /    /30\\\                 /    /30\\                 /    /30\\\
|         |                     |atan|--||                 |atan|--|||                 |atan|--||                 |atan|--||| |         |                     |atan|--||                 |atan|--|||                 |atan|--||                 |atan|--|||
|         |         4 ______    |    \71/|     4 ______    |    \71/||     4 ______    |    \71/|     4 ______    |    \71/|| |         |         4 ______    |    \71/|     4 ______    |    \71/||     4 ______    |    \71/|     4 ______    |    \71/||
|         |       5*\/ 5941 *sin|--------|   3*\/ 5941 *cos|--------||   3*\/ 5941 *sin|--------|   5*\/ 5941 *cos|--------|| |         |       3*\/ 5941 *cos|--------|   5*\/ 5941 *sin|--------||   5*\/ 5941 *cos|--------|   3*\/ 5941 *sin|--------||
|  7      |  11                 \   2    /                 \   2    /|                 \   2    /                 \   2    /| |  7      |  11                 \   2    /                 \   2    /|                 \   2    /                 \   2    /|
|- -- + I*|- -- - ------------------------ + ------------------------| + ------------------------ + ------------------------|*|- -- + I*|- -- - ------------------------ + ------------------------| - ------------------------ - ------------------------|
\  34     \  34              34                         34           /              34                         34           / \  34     \  34              34                         34           /              34                         34           /
(−734+359414sin⁡(atan⁡(3071)2)34+559414cos⁡(atan⁡(3071)2)34+i(−1134−559414sin⁡(atan⁡(3071)2)34+359414cos⁡(atan⁡(3071)2)34))(−559414cos⁡(atan⁡(3071)2)34−734−359414sin⁡(atan⁡(3071)2)34+i(−359414cos⁡(atan⁡(3071)2)34−1134+559414sin⁡(atan⁡(3071)2)34))\left(- \frac{7}{34} + \frac{3 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{5 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + i \left(- \frac{11}{34} - \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right)\right) \left(- \frac{5 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{7}{34} - \frac{3 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} + i \left(- \frac{3 \sqrt[4]{5941} \cos{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34} - \frac{11}{34} + \frac{5 \sqrt[4]{5941} \sin{\left(\frac{\operatorname{atan}{\left(\frac{30}{71} \right)}}{2} \right)}}{34}\right)\right) 
=
  31   22*I
- -- - ----
  17    17
−3117−22i17- \frac{31}{17} - \frac{22 i}{17} 
-31/17 - 22*i/17
Respuesta numérica [src] 
x1 = 1.21331499493745 + 0.17933780478976*i
x2 = -1.6250797008198 - 0.826396628319172*i
x2 = -1.6250797008198 - 0.826396628319172*i
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