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Ecuación x^2-14*x+33=0
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Expresiones idénticas
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- (2x-3)(3x+1)+2(x-5)(x+5)=2(1-2y)2+6x
- 2x-33x+1+2x-5x+5=21-2y2+6x
- (2x-3)(3x+1)+2(x-5)(x+5)=2(1-2y)²+6x
- (2x-3)(3x+1)+2(x-5)(x+5)=2(1-2y) en el grado 2+6x
- 2x-33x+1+2x-5x+5=21-2y^2+6x
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Expresiones semejantes
- (2x-3)(3x+1)+2(x-5)(x+5)=2(1-2y)^2-6x
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- (2x+3)(3x+1)+2(x-5)(x+5)=2(1-2y)^2+6x
- (2x-3)(3x+1)+2(x+5)(x+5)=2(1-2y)^2+6x
- (2x-3)(3x-1)+2(x-5)(x+5)=2(1-2y)^2+6x
- (2x-3)(3x+1)+2(x-5)(x-5)=2(1-2y)^2+6x
- (2x-3)(3x+1)+2(x-5)(x+5)=2(1+2y)^2+6x
(2x-3)(3x+1)+2(x-5)(x+5)=2(1-2y)^2+6x la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Ha introducido
[src]
2 (2*x - 3)*(3*x + 1) + 2*(x - 5)*(x + 5) = 2*(1 - 2*y) + 6*x
$$2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right) = 6 x + 2 \left(1 - 2 y\right)^{2}$$
Solución detallada
Transportemos el miembro derecho de la ecuación al
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right) = 6 x + 2 \left(1 - 2 y\right)^{2}$$
en
$$\left(- 6 x - 2 \left(1 - 2 y\right)^{2}\right) + \left(2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right)\right) = 0$$
Abramos la expresión en la ecuación
$$\left(- 6 x - 2 \left(1 - 2 y\right)^{2}\right) + \left(2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right)\right) = 0$$
Obtenemos la ecuación cuadrática
$$8 x^{2} - 13 x - 8 y^{2} + 8 y - 55 = 0$$
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
$$a = 8$$
$$b = -13$$
$$c = - 8 y^{2} + 8 y - 55$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{\sqrt{256 y^{2} - 256 y + 1929}}{16} + \frac{13}{16}$$
$$x_{2} = \frac{13}{16} - \frac{\sqrt{256 y^{2} - 256 y + 1929}}{16}$$
miembro izquierdo de la ecuación con el signo negativo.
La ecuación se convierte de
$$2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right) = 6 x + 2 \left(1 - 2 y\right)^{2}$$
en
$$\left(- 6 x - 2 \left(1 - 2 y\right)^{2}\right) + \left(2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right)\right) = 0$$
Abramos la expresión en la ecuación
$$\left(- 6 x - 2 \left(1 - 2 y\right)^{2}\right) + \left(2 \left(x - 5\right) \left(x + 5\right) + \left(2 x - 3\right) \left(3 x + 1\right)\right) = 0$$
Obtenemos la ecuación cuadrática
$$8 x^{2} - 13 x - 8 y^{2} + 8 y - 55 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
donde D = b^2 - 4*a*c es el discriminante.
Como
$$a = 8$$
$$b = -13$$
$$c = - 8 y^{2} + 8 y - 55$$
, entonces
D = b^2 - 4 * a * c =
(-13)^2 - 4 * (8) * (-55 - 8*y^2 + 8*y) = 1929 - 256*y + 256*y^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{\sqrt{256 y^{2} - 256 y + 1929}}{16} + \frac{13}{16}$$
$$x_{2} = \frac{13}{16} - \frac{\sqrt{256 y^{2} - 256 y + 1929}}{16}$$
Gráfica
Suma y producto de raíces
[src]
suma
_________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________ _________________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/|
\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *cos|-------------------------------------------------------------------------------| I*\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *sin|-------------------------------------------------------------------------------| \/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *cos|-------------------------------------------------------------------------------| I*\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *sin|-------------------------------------------------------------------------------|
13 \ 2 / \ 2 / 13 \ 2 / \ 2 /
-- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + -- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16 16 16 16 16 16
$$\left(- \frac{i \sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} - \frac{\sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{13}{16}\right) + \left(\frac{i \sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{\sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{13}{16}\right)$$
=
13/8
$$\frac{13}{8}$$
producto
/ _________________________________________________________________________________ _________________________________________________________________________________ \ / _________________________________________________________________________________ _________________________________________________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/|| | 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/|| | \/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *cos|-------------------------------------------------------------------------------| I*\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *sin|-------------------------------------------------------------------------------|| | \/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *cos|-------------------------------------------------------------------------------| I*\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *sin|-------------------------------------------------------------------------------|| |13 \ 2 / \ 2 /| |13 \ 2 / \ 2 /| |-- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------|*|-- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------| \16 16 16 / \16 16 16 /
$$\left(- \frac{i \sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} - \frac{\sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{13}{16}\right) \left(\frac{i \sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{\sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{13}{16}\right)$$
=
55 2 2 - -- + im (y) - re (y) + I*im(y) - 2*I*im(y)*re(y) + re(y) 8
$$- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + \operatorname{re}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2} + i \operatorname{im}{\left(y\right)} - \frac{55}{8}$$
-55/8 + im(y)^2 - re(y)^2 + i*im(y) - 2*i*im(y)*re(y) + re(y)
Respuesta rápida
[src]
_________________________________________________________________________________ _________________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/|
\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *cos|-------------------------------------------------------------------------------| I*\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *sin|-------------------------------------------------------------------------------|
13 \ 2 / \ 2 /
x1 = -- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- - ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16 16 16
$$x_{1} = - \frac{i \sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} - \frac{\sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{13}{16}$$
_________________________________________________________________________________ _________________________________________________________________________________
/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/| 4 / 2 / 2 2 \ |atan2\-256*im(y) + 512*im(y)*re(y), 1929 - 256*im (y) - 256*re(y) + 256*re (y)/|
\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *cos|-------------------------------------------------------------------------------| I*\/ (-256*im(y) + 512*im(y)*re(y)) + \1929 - 256*im (y) - 256*re(y) + 256*re (y)/ *sin|-------------------------------------------------------------------------------|
13 \ 2 / \ 2 /
x2 = -- + -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------
16 16 16
$$x_{2} = \frac{i \sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{\sqrt[4]{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)}\right)^{2} + \left(256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(512 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} - 256 \operatorname{im}{\left(y\right)},256 \left(\operatorname{re}{\left(y\right)}\right)^{2} - 256 \operatorname{re}{\left(y\right)} - 256 \left(\operatorname{im}{\left(y\right)}\right)^{2} + 1929 \right)}}{2} \right)}}{16} + \frac{13}{16}$$
x2 = i*((512*re(y)*im(y) - 256*im(y))^2 + (256*re(y)^2 - 256*re(y) - 256*im(y)^2 + 1929)^2)^(1/4)*sin(atan2(512*re(y)*im(y) - 256*im(y, 256*re(y)^2 - 256*re(y) - 256*im(y)^2 + 1929)/2)/16 + ((512*re(y)*im(y) - 256*im(y))^2 + (256*re(y)^2 - 256*re(y) - 256*im(y)^2 + 1929)^2)^(1/4)*cos(atan2(512*re(y)*im(y) - 256*im(y), 256*re(y)^2 - 256*re(y) - 256*im(y)^2 + 1929)/2)/16 + 13/16)