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- La ecuación:
-
Ecuación x+9=0
-
Ecuación 3*x^2=27
-
Ecuación (x-2)/(x+2)=(x+3)/(x-4)
-
Ecuación x^2-4=0
- Expresar {x} en función de y en la ecuación:
- -6*x+11*y=-5
- -15*x+7*y=1
- -11*x-11*y=4
- -14*x+4*y=-16
- Integral de d{x}:
- sqrt(x)
- x+a
- Derivada de:
-
sqrt(x)
- Límite de la función:
-
sqrt(x)
-
Expresiones idénticas
- sqrt(x)=x+a
- raíz cuadrada de (x) es igual a x más a
- √(x)=x+a
- sqrtx=x+a
-
Expresiones semejantes
- 3sqrt(x^2-2*x)=0
- sqrt(x)=x-a
- 4*asin(x)+acos(x)=pi
- y=x*arctan(sqrt(x^2-1))
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(-14+9*x)=x
- sqrt(x+a)=a-sqrtx
- sqrt(x^2+2x+5)=sqrt(x^2+3x+10)
- sqrt(1+4*x-sqr(x))=x-1
- sqrt(x+2)-4=0
sqrt(x)=x+a la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$\sqrt{x} = a + x$$
$$\sqrt{x} = a + x$$
Elevemos las dos partes de la ecuación a la potencia 2
$$x = \left(a + x\right)^{2}$$
$$x = a^{2} + 2 a x + x^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- a^{2} - 2 a x - x^{2} + x = 0$$
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
$$a = -1$$
$$b = 1 - 2 a$$
$$c = - a^{2}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = - a - \frac{\sqrt{- 4 a^{2} + \left(1 - 2 a\right)^{2}}}{2} + \frac{1}{2}$$
$$x_{2} = - a + \frac{\sqrt{- 4 a^{2} + \left(1 - 2 a\right)^{2}}}{2} + \frac{1}{2}$$
$$\sqrt{x} = a + x$$
$$\sqrt{x} = a + x$$
Elevemos las dos partes de la ecuación a la potencia 2
$$x = \left(a + x\right)^{2}$$
$$x = a^{2} + 2 a x + x^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- a^{2} - 2 a x - x^{2} + x = 0$$
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
$$a = -1$$
$$b = 1 - 2 a$$
$$c = - a^{2}$$
, entonces
D = b^2 - 4 * a * c =
(1 - 2*a)^2 - 4 * (-1) * (-a^2) = (1 - 2*a)^2 - 4*a^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - a - \frac{\sqrt{- 4 a^{2} + \left(1 - 2 a\right)^{2}}}{2} + \frac{1}{2}$$
$$x_{2} = - a + \frac{\sqrt{- 4 a^{2} + \left(1 - 2 a\right)^{2}}}{2} + \frac{1}{2}$$
Gráfica
Respuesta rápida
[src]
/ ____________________________ \ ____________________________
| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\
| \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------|
1 | \ 2 /| \ 2 /
x1 = - - re(a) + I*|-im(a) - -----------------------------------------------------------------| - -----------------------------------------------------------------
2 \ 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}$$
/ ____________________________ \ ____________________________
| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\
| \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------|
1 | \ 2 /| \ 2 /
x2 = - - re(a) + I*|-im(a) + -----------------------------------------------------------------| + -----------------------------------------------------------------
2 \ 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}$$
x2 = i*(((1 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*sin(atan2(-4*im(a, 1 - 4*re(a))/2)/2 - im(a)) + ((1 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*cos(atan2(-4*im(a), 1 - 4*re(a))/2)/2 - re(a) + 1/2)
Suma y producto de raíces
[src]
suma
/ ____________________________ \ ____________________________ / ____________________________ \ ____________________________
| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ | 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\
| \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| | \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------|
1 | \ 2 /| \ 2 / 1 | \ 2 /| \ 2 /
- - re(a) + I*|-im(a) - -----------------------------------------------------------------| - ----------------------------------------------------------------- + - - re(a) + I*|-im(a) + -----------------------------------------------------------------| + -----------------------------------------------------------------
2 \ 2 / 2 2 \ 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right)$$
=
/ ____________________________ \ / ____________________________ \
| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\|
| \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| | \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------||
| \ 2 /| | \ 2 /|
1 - 2*re(a) + I*|-im(a) + -----------------------------------------------------------------| + I*|-im(a) - -----------------------------------------------------------------|
\ 2 / \ 2 /
$$i \left(- \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + i \left(\frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - 2 \operatorname{re}{\left(a\right)} + 1$$
producto
/ / ____________________________ \ ____________________________ \ / / ____________________________ \ ____________________________ \ | | 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| | | 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| | | \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------|| | | \/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------|| |1 | \ 2 /| \ 2 /| |1 | \ 2 /| \ 2 /| |- - re(a) + I*|-im(a) - -----------------------------------------------------------------| - -----------------------------------------------------------------|*|- - re(a) + I*|-im(a) + -----------------------------------------------------------------| + -----------------------------------------------------------------| \2 \ 2 / 2 / \2 \ 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{im}{\left(a\right)}\right) + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \operatorname{re}{\left(a\right)} + \frac{1}{2}\right)$$
=
2 2 re (a) - im (a) + 2*I*im(a)*re(a)
$$\left(\operatorname{re}{\left(a\right)}\right)^{2} + 2 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - \left(\operatorname{im}{\left(a\right)}\right)^{2}$$
re(a)^2 - im(a)^2 + 2*i*im(a)*re(a)