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- La ecuación:
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Ecuación 6*x^2+x-7=0
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- Expresar {x} en función de y en la ecuación:
- 6*x+4*y=15
- -4*x+4*y=-9
- 3*x-12*y=-14
- 1*x+4*y=-16
- Integral de d{x}:
- sqrt(8-x^2)
- a+x
- Gráfico de la función y =:
-
sqrt(8-x^2)
- Expresión:
- a+x
-
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- sqrt(8-x2)=a+x
- sqrt8-x2=a+x
- sqrt(8-x²)=a+x
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- sqrt8-x^2=a+x
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Expresiones semejantes
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- x^4+2x^3(1+a)+x^2(4a-1)-2x(1+a)-4a=0
-
Expresiones con funciones
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- sqrt(x-6)=sqrt(4-x)
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sqrt(8-x^2)=a+x la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$\sqrt{8 - x^{2}} = a + x$$
$$\sqrt{8 - x^{2}} = a + x$$
Elevemos las dos partes de la ecuación a la potencia 2
$$8 - x^{2} = \left(a + x\right)^{2}$$
$$8 - x^{2} = a^{2} + 2 a x + x^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- a^{2} - 2 a x - 2 x^{2} + 8 = 0$$
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
$$a = -2$$
$$b = - 2 a$$
$$c = 8 - a^{2}$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = - \frac{a}{2} - \frac{\sqrt{64 - 4 a^{2}}}{4}$$
$$x_{2} = - \frac{a}{2} + \frac{\sqrt{64 - 4 a^{2}}}{4}$$
$$\sqrt{8 - x^{2}} = a + x$$
$$\sqrt{8 - x^{2}} = a + x$$
Elevemos las dos partes de la ecuación a la potencia 2
$$8 - x^{2} = \left(a + x\right)^{2}$$
$$8 - x^{2} = a^{2} + 2 a x + x^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- a^{2} - 2 a x - 2 x^{2} + 8 = 0$$
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
$$a = -2$$
$$b = - 2 a$$
$$c = 8 - a^{2}$$
, entonces
D = b^2 - 4 * a * c =
(-2*a)^2 - 4 * (-2) * (8 - a^2) = 64 - 4*a^2
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = - \frac{a}{2} - \frac{\sqrt{64 - 4 a^{2}}}{4}$$
$$x_{2} = - \frac{a}{2} + \frac{\sqrt{64 - 4 a^{2}}}{4}$$
Gráfica
Respuesta rápida
[src]
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|
| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *cos|-------------------------------------------|
re(a) | im(a) \ 2 /| \ 2 /
x1 = - ----- + I*|- ----- - ------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(a\right)}}{2}$$
/ ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|
| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *cos|-------------------------------------------|
re(a) | im(a) \ 2 /| \ 2 /
x2 = - ----- + I*|- ----- + ------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------
2 \ 2 2 / 2
$$x_{2} = i \left(\frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(a\right)}}{2}$$
x2 = i*(((-re(a)^2 + im(a)^2 + 16)^2 + 4*re(a)^2*im(a)^2)^(1/4)*sin(atan2(-2*re(a)*im(a, -re(a)^2 + im(a)^2 + 16)/2)/2 - im(a)/2) + ((-re(a)^2 + im(a)^2 + 16)^2 + 4*re(a)^2*im(a)^2)^(1/4)*cos(atan2(-2*re(a)*im(a), -re(a)^2 + im(a)^2 + 16)/2)/2 - re(a)/2)
Suma y producto de raíces
[src]
suma
/ ___________________________________________ \ ___________________________________________ / ___________________________________________ \ ___________________________________________
| / 2 / / 2 2 \\| / 2 / / 2 2 \\ | / 2 / / 2 2 \\| / 2 / / 2 2 \\
| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/| | 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|
| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *cos|-------------------------------------------| | \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *cos|-------------------------------------------|
re(a) | im(a) \ 2 /| \ 2 / re(a) | im(a) \ 2 /| \ 2 /
- ----- + I*|- ----- - ------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------ + - ----- + I*|- ----- + ------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(a\right)}}{2}\right) + \left(i \left(\frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(a\right)}}{2}\right)$$
=
/ ___________________________________________ \ / ___________________________________________ \
| / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| | 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/||
| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| | \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------||
| im(a) \ 2 /| | im(a) \ 2 /|
-re(a) + I*|- ----- + ------------------------------------------------------------------------------------------------| + I*|- ----- - ------------------------------------------------------------------------------------------------|
\ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \operatorname{re}{\left(a\right)}$$
producto
/ / ___________________________________________ \ ___________________________________________ \ / / ___________________________________________ \ ___________________________________________ \ | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| / 2 / / 2 2 \\| | | 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| | | 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| 4 / / 2 2 \ 2 2 |atan2\-2*im(a)*re(a), 16 + im (a) - re (a)/|| | | \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *cos|-------------------------------------------|| | | \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *sin|-------------------------------------------|| \/ \16 + im (a) - re (a)/ + 4*im (a)*re (a) *cos|-------------------------------------------|| | re(a) | im(a) \ 2 /| \ 2 /| | re(a) | im(a) \ 2 /| \ 2 /| |- ----- + I*|- ----- - ------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------|*|- ----- + I*|- ----- + ------------------------------------------------------------------------------------------------| + ------------------------------------------------------------------------------------------------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) - \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(a\right)}}{2}\right) \left(i \left(\frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{im}{\left(a\right)}}{2}\right) + \frac{\sqrt[4]{\left(- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16\right)^{2} + 4 \left(\operatorname{re}{\left(a\right)}\right)^{2} \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)},- \left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 16 \right)}}{2} \right)}}{2} - \frac{\operatorname{re}{\left(a\right)}}{2}\right)$$
=
2 2
re (a) im (a)
-4 + ------ - ------ + I*im(a)*re(a)
2 2
$$\frac{\left(\operatorname{re}{\left(a\right)}\right)^{2}}{2} + i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} - \frac{\left(\operatorname{im}{\left(a\right)}\right)^{2}}{2} - 4$$
-4 + re(a)^2/2 - im(a)^2/2 + i*im(a)*re(a)