### Other calculators # x^4-x^3+5*x^2=0 equation

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#### Numerical solution:

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### The solution

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 4    3      2
x  - x  + 5*x  = 0
$$5 x^{2} + \left(x^{4} - x^{3}\right) = 0$$
Detail solution
Given the equation:
$$5 x^{2} + \left(x^{4} - x^{3}\right) = 0$$
transform
Take common factor x from the equation
we get:
$$x \left(x^{2} - x + 5\right) = 0$$
then:
$$x_{1} = 0$$
and also
we get the equation
$$x^{2} - x + 5 = 0$$
This equation is of the form
a*x^2 + b*x + c = 0

A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{2} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{3} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -1$$
$$c = 5$$
, then
D = b^2 - 4 * a * c =

(-1)^2 - 4 * (1) * (5) = -19

Because D<0, then the equation
has no real roots,
but complex roots is exists.
x2 = (-b + sqrt(D)) / (2*a)

x3 = (-b - sqrt(D)) / (2*a)

or
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
The final answer for x^4 - x^3 + 5*x^2 = 0:
$$x_{1} = 0$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
The graph
Sum and product of roots [src]
sum
        ____           ____
1   I*\/ 19    1   I*\/ 19
- - -------- + - + --------
2      2       2      2    
$$\left(\frac{1}{2} - \frac{\sqrt{19} i}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
1
$$1$$
product
  /        ____\ /        ____\
|1   I*\/ 19 | |1   I*\/ 19 |
0*|- - --------|*|- + --------|
\2      2    / \2      2    /
$$0 \left(\frac{1}{2} - \frac{\sqrt{19} i}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{19} i}{2}\right)$$
=
0
$$0$$
0
Rapid solution [src]
x1 = 0
$$x_{1} = 0$$
             ____
1   I*\/ 19
x2 = - - --------
2      2    
$$x_{2} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$
             ____
1   I*\/ 19
x3 = - + --------
2      2    
$$x_{3} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
x3 = 1/2 + sqrt(19)*i/2
x1 = 0.5 - 2.17944947177034*i
x2 = 0.5 + 2.17944947177034*i
x3 = 0.0
x3 = 0.0