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- La ecuación:
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Ecuación x^2+5x=36
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Ecuación 3^x=-1
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Ecuación 9^x-8*3^x-9=0
- Expresar {x} en función de y en la ecuación:
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Expresiones idénticas
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- 9x2-3x+c=0
- 9x²-3x+c=0
- 9x en el grado 2-3x+c=0
- 9x^2-3x+c=O
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Expresiones semejantes
- 9x^2-3x-c=0
- 9x^2+3x+c=0
9x^2-3x+c=0 la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Es la ecuación de la forma
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 9$$
$$b = -3$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{9 - 36 c}}{18} + \frac{1}{6}$$
$$x_{2} = \frac{1}{6} - \frac{\sqrt{9 - 36 c}}{18}$$
a*x^2 + b*x + c = 0
La ecuación cuadrática puede ser resuelta
con la ayuda del discriminante.
Las raíces de la ecuación cuadrática:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 9$$
$$b = -3$$
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
$$x_{1} = \frac{\sqrt{9 - 36 c}}{18} + \frac{1}{6}$$
$$x_{2} = \frac{1}{6} - \frac{\sqrt{9 - 36 c}}{18}$$
Teorema de Cardano-Vieta
reescribamos la ecuación
$$c + \left(9 x^{2} - 3 x\right) = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{c}{9} + x^{2} - \frac{x}{3} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{1}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{9}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{3}$$
$$x_{1} x_{2} = \frac{c}{9}$$
$$c + \left(9 x^{2} - 3 x\right) = 0$$
de
$$a x^{2} + b x + c = 0$$
como ecuación cuadrática reducida
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{c}{9} + x^{2} - \frac{x}{3} = 0$$
$$p x + q + x^{2} = 0$$
donde
$$p = \frac{b}{a}$$
$$p = - \frac{1}{3}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{9}$$
Fórmulas de Cardano-Vieta
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{1}{3}$$
$$x_{1} x_{2} = \frac{c}{9}$$
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
$$\left(- \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right) + \left(\frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right)$$
=
1/3
$$\frac{1}{3}$$
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 /
$$\left(- \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right) \left(\frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}\right)$$
=
re(c) I*im(c) ----- + ------- 9 9
$$\frac{\operatorname{re}{\left(c\right)}}{9} + \frac{i \operatorname{im}{\left(c\right)}}{9}$$
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
$$x_{1} = - \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}$$
____________________________ ____________________________
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
$$x_{2} = \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(c\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(c\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(c\right)},1 - 4 \operatorname{re}{\left(c\right)} \right)}}{2} \right)}}{6} + \frac{1}{6}$$
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)