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sqrt(3x-5b)=5b-2x la ecuación

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Solución numérica:

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Solución

Ha introducido [src] 
  ___________            
\/ 3*x - 5*b  = 5*b - 2*x
−5b+3x=5b−2x\sqrt{- 5 b + 3 x} = 5 b - 2 x
Simplificar
Solución detallada
Tenemos la ecuación
−5b+3x=5b−2x\sqrt{- 5 b + 3 x} = 5 b - 2 x
−5b+3x=5b−2x\sqrt{- 5 b + 3 x} = 5 b - 2 x
Elevemos las dos partes de la ecuación a la potencia 2
−5b+3x=(5b−2x)2- 5 b + 3 x = \left(5 b - 2 x\right)^{2}
−5b+3x=25b2−20bx+4x2- 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
−25b2+20bx−5b−4x2+3x=0- 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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
donde D = b^2 - 4*a*c es el discriminante.
Como
a=−4a = -4
b=20b+3b = 20 b + 3
c=−25b2−5bc = - 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
x1=5b2−−400b2−80b+(20b+3)28+38x_{1} = \frac{5 b}{2} - \frac{\sqrt{- 400 b^{2} - 80 b + \left(20 b + 3\right)^{2}}}{8} + \frac{3}{8}
x2=5b2+−400b2−80b+(20b+3)28+38x_{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
(i(−(40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)−(40re⁡(b)+9)2+1600(im⁡(b))24cos⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5re⁡(b)2+38)+(i((40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)+(40re⁡(b)+9)2+1600(im⁡(b))24cos⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5re⁡(b)2+38)\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(−(40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)+i((40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)+5re⁡(b)+34i \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                                  /
(i(−(40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)−(40re⁡(b)+9)2+1600(im⁡(b))24cos⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5re⁡(b)2+38)(i((40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)+(40re⁡(b)+9)2+1600(im⁡(b))24cos⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5re⁡(b)2+38)\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
25(re⁡(b))24+25ire⁡(b)im⁡(b)2+5re⁡(b)4−25(im⁡(b))24+5iim⁡(b)4\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
x1=i(−(40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)−(40re⁡(b)+9)2+1600(im⁡(b))24cos⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5re⁡(b)2+38x_{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
x2=i((40re⁡(b)+9)2+1600(im⁡(b))24sin⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5im⁡(b)2)+(40re⁡(b)+9)2+1600(im⁡(b))24cos⁡(atan2⁡(40im⁡(b),40re⁡(b)+9)2)8+5re⁡(b)2+38x_{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)
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