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- La ecuación:
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Ecuación -x+7=0
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Ecuación x+(x/4)=-5
- Ecuación z^4-81=0
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Ecuación z^3=8
- Expresar {x} en función de y en la ecuación:
- -18*x-8*y=-18
- -13*x+12*y=-19
- -1*x+11*y=-9
- 16*x-11*y=13
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Expresiones idénticas
- sqrt(3x-5b)=5b-2x
- raíz cuadrada de (3x menos 5b) es igual a 5b menos 2x
- √(3x-5b)=5b-2x
- sqrt3x-5b=5b-2x
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Expresiones semejantes
- sqrt(3x+5b)=5b-2x
- sqrt(3x-5b)=5b+2x
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Expresiones con funciones
- Raíz cuadrada sqrt
- sqrt(4*sin(x)^2+1)+sqrt(4*cos(x)^2+3)=4
- sqrt(x-2)*sqrt(x+2)=sqrt(8)
- sqrt(5x)=2x(1/2)
- sqrt5(x−23)=−3
- sqrt(3)*x/sqrt(x^2-8*x-64)=383/1000
sqrt(3x-5b)=5b-2x la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Tenemos la ecuación
$$\sqrt{- 5 b + 3 x} = 5 b - 2 x$$
$$\sqrt{- 5 b + 3 x} = 5 b - 2 x$$
Elevemos las dos partes de la ecuación a la potencia 2
$$- 5 b + 3 x = \left(5 b - 2 x\right)^{2}$$
$$- 5 b + 3 x = 25 b^{2} - 20 b x + 4 x^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- 25 b^{2} + 20 b x - 5 b - 4 x^{2} + 3 x = 0$$
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
$$a = -4$$
$$b = 20 b + 3$$
$$c = - 25 b^{2} - 5 b$$
, entonces
La ecuación tiene dos raíces.
o
$$x_{1} = \frac{5 b}{2} - \frac{\sqrt{- 400 b^{2} - 80 b + \left(20 b + 3\right)^{2}}}{8} + \frac{3}{8}$$
$$x_{2} = \frac{5 b}{2} + \frac{\sqrt{- 400 b^{2} - 80 b + \left(20 b + 3\right)^{2}}}{8} + \frac{3}{8}$$
$$\sqrt{- 5 b + 3 x} = 5 b - 2 x$$
$$\sqrt{- 5 b + 3 x} = 5 b - 2 x$$
Elevemos las dos partes de la ecuación a la potencia 2
$$- 5 b + 3 x = \left(5 b - 2 x\right)^{2}$$
$$- 5 b + 3 x = 25 b^{2} - 20 b x + 4 x^{2}$$
Transpongamos la parte derecha de la ecuación miembro izquierdo de la ecuación con el signo negativo
$$- 25 b^{2} + 20 b x - 5 b - 4 x^{2} + 3 x = 0$$
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
$$a = -4$$
$$b = 20 b + 3$$
$$c = - 25 b^{2} - 5 b$$
, entonces
D = b^2 - 4 * a * c =
(3 + 20*b)^2 - 4 * (-4) * (-25*b^2 - 5*b) = (3 + 20*b)^2 - 400*b^2 - 80*b
La ecuación tiene dos raíces.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
o
$$x_{1} = \frac{5 b}{2} - \frac{\sqrt{- 400 b^{2} - 80 b + \left(20 b + 3\right)^{2}}}{8} + \frac{3}{8}$$
$$x_{2} = \frac{5 b}{2} + \frac{\sqrt{- 400 b^{2} - 80 b + \left(20 b + 3\right)^{2}}}{8} + \frac{3}{8}$$
Gráfica
Suma y producto de raíces
[src]
suma
/ _______________________________ \ _______________________________ / _______________________________ \ _______________________________
| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\ | 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\
| \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| \/ (9 + 40*re(b)) + 1600*im (b) *cos|-----------------------------| | \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| \/ (9 + 40*re(b)) + 1600*im (b) *cos|-----------------------------|
3 5*re(b) |5*im(b) \ 2 /| \ 2 / 3 5*re(b) |5*im(b) \ 2 /| \ 2 /
- + ------- + I*|------- - ---------------------------------------------------------------------| - --------------------------------------------------------------------- + - + ------- + I*|------- + ---------------------------------------------------------------------| + ---------------------------------------------------------------------
8 2 \ 2 8 / 8 8 2 \ 2 8 / 8
$$\left(i \left(- \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) - \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{re}{\left(b\right)}}{2} + \frac{3}{8}\right) + \left(i \left(\frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) + \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{re}{\left(b\right)}}{2} + \frac{3}{8}\right)$$
=
/ _______________________________ \ / _______________________________ \
| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| | 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\|
| \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| | \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------||
3 |5*im(b) \ 2 /| |5*im(b) \ 2 /|
- + 5*re(b) + I*|------- - ---------------------------------------------------------------------| + I*|------- + ---------------------------------------------------------------------|
4 \ 2 8 / \ 2 8 /
$$i \left(- \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) + i \left(\frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) + 5 \operatorname{re}{\left(b\right)} + \frac{3}{4}$$
producto
/ / _______________________________ \ _______________________________ \ / / _______________________________ \ _______________________________ \ | | 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| | | 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| | | \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| \/ (9 + 40*re(b)) + 1600*im (b) *cos|-----------------------------|| | | \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| \/ (9 + 40*re(b)) + 1600*im (b) *cos|-----------------------------|| |3 5*re(b) |5*im(b) \ 2 /| \ 2 /| |3 5*re(b) |5*im(b) \ 2 /| \ 2 /| |- + ------- + I*|------- - ---------------------------------------------------------------------| - ---------------------------------------------------------------------|*|- + ------- + I*|------- + ---------------------------------------------------------------------| + ---------------------------------------------------------------------| \8 2 \ 2 8 / 8 / \8 2 \ 2 8 / 8 /
$$\left(i \left(- \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) - \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{re}{\left(b\right)}}{2} + \frac{3}{8}\right) \left(i \left(\frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) + \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{re}{\left(b\right)}}{2} + \frac{3}{8}\right)$$
=
2 2
25*im (b) 5*re(b) 25*re (b) 5*I*im(b) 25*I*im(b)*re(b)
- --------- + ------- + --------- + --------- + ----------------
4 4 4 4 2
$$\frac{25 \left(\operatorname{re}{\left(b\right)}\right)^{2}}{4} + \frac{25 i \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)}}{2} + \frac{5 \operatorname{re}{\left(b\right)}}{4} - \frac{25 \left(\operatorname{im}{\left(b\right)}\right)^{2}}{4} + \frac{5 i \operatorname{im}{\left(b\right)}}{4}$$
-25*im(b)^2/4 + 5*re(b)/4 + 25*re(b)^2/4 + 5*i*im(b)/4 + 25*i*im(b)*re(b)/2
Respuesta rápida
[src]
/ _______________________________ \ _______________________________
| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\
| \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| \/ (9 + 40*re(b)) + 1600*im (b) *cos|-----------------------------|
3 5*re(b) |5*im(b) \ 2 /| \ 2 /
x1 = - + ------- + I*|------- - ---------------------------------------------------------------------| - ---------------------------------------------------------------------
8 2 \ 2 8 / 8
$$x_{1} = i \left(- \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) - \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{re}{\left(b\right)}}{2} + \frac{3}{8}$$
/ _______________________________ \ _______________________________
| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\| 4 / 2 2 /atan2(40*im(b), 9 + 40*re(b))\
| \/ (9 + 40*re(b)) + 1600*im (b) *sin|-----------------------------|| \/ (9 + 40*re(b)) + 1600*im (b) *cos|-----------------------------|
3 5*re(b) |5*im(b) \ 2 /| \ 2 /
x2 = - + ------- + I*|------- + ---------------------------------------------------------------------| + ---------------------------------------------------------------------
8 2 \ 2 8 / 8
$$x_{2} = i \left(\frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{im}{\left(b\right)}}{2}\right) + \frac{\sqrt[4]{\left(40 \operatorname{re}{\left(b\right)} + 9\right)^{2} + 1600 \left(\operatorname{im}{\left(b\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(40 \operatorname{im}{\left(b\right)},40 \operatorname{re}{\left(b\right)} + 9 \right)}}{2} \right)}}{8} + \frac{5 \operatorname{re}{\left(b\right)}}{2} + \frac{3}{8}$$
x2 = i*(((40*re(b) + 9)^2 + 1600*im(b)^2)^(1/4)*sin(atan2(40*im(b, 40*re(b) + 9)/2)/8 + 5*im(b)/2) + ((40*re(b) + 9)^2 + 1600*im(b)^2)^(1/4)*cos(atan2(40*im(b), 40*re(b) + 9)/2)/8 + 5*re(b)/2 + 3/8)