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9x^2-3x+c=0 la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
   2              
9*x  - 3*x + c = 0
c+(9x2−3x)=0c + \left(9 x^{2} - 3 x\right) = 0
Simplificar
Solución detallada
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=9a = 9
b=−3b = -3
True

, entonces
D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (9) * (c) = 9 - 36*c

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

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

o
x1=9−36c18+16x_{1} = \frac{\sqrt{9 - 36 c}}{18} + \frac{1}{6}
x2=16−9−36c18x_{2} = \frac{1}{6} - \frac{\sqrt{9 - 36 c}}{18}
Teorema de Cardano-Vieta
reescribamos la ecuación
c+(9x2−3x)=0c + \left(9 x^{2} - 3 x\right) = 0
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
c9+x2−x3=0\frac{c}{9} + x^{2} - \frac{x}{3} = 0
px+q+x2=0p x + q + x^{2} = 0
donde
p=bap = \frac{b}{a}
p=−13p = - \frac{1}{3}
q=caq = \frac{c}{a}
q=c9q = \frac{c}{9}
Fórmulas de Cardano-Vieta
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=13x_{1} + x_{2} = \frac{1}{3}
x1x2=c9x_{1} x_{2} = \frac{c}{9}
Gráfica
Suma y producto de raíces [src] 
suma
       ____________________________                                          ____________________________                                            ____________________________                                          ____________________________                                  
    4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\     4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\       4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\     4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\
    \/  (1 - 4*re(c))  + 16*im (c) *cos|----------------------------|   I*\/  (1 - 4*re(c))  + 16*im (c) *sin|----------------------------|       \/  (1 - 4*re(c))  + 16*im (c) *cos|----------------------------|   I*\/  (1 - 4*re(c))  + 16*im (c) *sin|----------------------------|
1                                      \             2              /                                        \             2              /   1                                      \             2              /                                        \             2              /
- - ----------------------------------------------------------------- - ------------------------------------------------------------------- + - + ----------------------------------------------------------------- + -------------------------------------------------------------------
6                                   6                                                                    6                                    6                                   6                                                                    6
(−i(1−4re⁡(c))2+16(im⁡(c))24sin⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6−(1−4re⁡(c))2+16(im⁡(c))24cos⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+16)+(i(1−4re⁡(c))2+16(im⁡(c))24sin⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+(1−4re⁡(c))2+16(im⁡(c))24cos⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+16)\left(- \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right) + \left(\frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right) 
=
1/3
13\frac{1}{3} 
producto
/       ____________________________                                          ____________________________                                  \ /       ____________________________                                          ____________________________                                  \
|    4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\     4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\| |    4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\     4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\|
|    \/  (1 - 4*re(c))  + 16*im (c) *cos|----------------------------|   I*\/  (1 - 4*re(c))  + 16*im (c) *sin|----------------------------|| |    \/  (1 - 4*re(c))  + 16*im (c) *cos|----------------------------|   I*\/  (1 - 4*re(c))  + 16*im (c) *sin|----------------------------||
|1                                      \             2              /                                        \             2              /| |1                                      \             2              /                                        \             2              /|
|- - ----------------------------------------------------------------- - -------------------------------------------------------------------|*|- + ----------------------------------------------------------------- + -------------------------------------------------------------------|
\6                                   6                                                                    6                                 / \6                                   6                                                                    6                                 /
(−i(1−4re⁡(c))2+16(im⁡(c))24sin⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6−(1−4re⁡(c))2+16(im⁡(c))24cos⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+16)(i(1−4re⁡(c))2+16(im⁡(c))24sin⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+(1−4re⁡(c))2+16(im⁡(c))24cos⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+16)\left(- \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right) \left(\frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right) 
=
re(c)   I*im(c)
----- + -------
  9        9
re⁡(c)9+iim⁡(c)9\frac{\operatorname{re}{\left(c\right)}}{9} + \frac{i \operatorname{im}{\left(c\right)}}{9} 
re(c)/9 + i*im(c)/9
Respuesta rápida [src] 
            ____________________________                                          ____________________________                                  
         4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\     4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\
         \/  (1 - 4*re(c))  + 16*im (c) *cos|----------------------------|   I*\/  (1 - 4*re(c))  + 16*im (c) *sin|----------------------------|
     1                                      \             2              /                                        \             2              /
x1 = - - ----------------------------------------------------------------- - -------------------------------------------------------------------
     6                                   6                                                                    6
x1=−i(1−4re⁡(c))2+16(im⁡(c))24sin⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6−(1−4re⁡(c))2+16(im⁡(c))24cos⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+16x_{1} = - \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6} 
            ____________________________                                          ____________________________                                  
         4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\     4 /              2        2        /atan2(-4*im(c), 1 - 4*re(c))\
         \/  (1 - 4*re(c))  + 16*im (c) *cos|----------------------------|   I*\/  (1 - 4*re(c))  + 16*im (c) *sin|----------------------------|
     1                                      \             2              /                                        \             2              /
x2 = - + ----------------------------------------------------------------- + -------------------------------------------------------------------
     6                                   6                                                                    6
x2=i(1−4re⁡(c))2+16(im⁡(c))24sin⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+(1−4re⁡(c))2+16(im⁡(c))24cos⁡(atan2⁡(−4im⁡(c),1−4re⁡(c))2)6+16x_{2} = \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6} 
x2 = i*((1 - 4*re(c))^2 + 16*im(c)^2)^(1/4)*sin(atan2(-4*im(c, 1 - 4*re(c))/2)/6 + ((1 - 4*re(c))^2 + 16*im(c)^2)^(1/4)*cos(atan2(-4*im(c), 1 - 4*re(c))/2)/6 + 1/6)
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