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:
$$x_{1} = 0$$
$$x_{2} = \frac{1}{2} + \frac{\sqrt{19} i}{2}$$
$$x_{3} = \frac{1}{2} - \frac{\sqrt{19} i}{2}$$