Given the equation:
$$x^{3} + 4 x^{2} - 9 x - 36 = 0$$
transform
$$\left(- 9 x - \left(- x^{3} - 4 x^{2} + 63\right)\right) + 27 = 0$$
or
$$\left(- 9 x - \left(- x^{3} - 4 x^{2} + 27 + 36\right)\right) + 3 \cdot 9 = 0$$
$$- 9 \left(x - 3\right) + \left(4 \left(x^{2} - 3^{2}\right) + 1 \left(x^{3} - 3^{3}\right)\right) = 0$$
$$- 9 \left(x - 3\right) + \left(1 \left(x - 3\right) \left(\left(x^{2} + 3 x\right) + 3^{2}\right) + 4 \left(x - 3\right) \left(x + 3\right)\right) = 0$$
Take common factor -3 + x from the equation
we get:
$$\left(x - 3\right) \left(\left(4 \left(x + 3\right) + 1 \left(\left(x^{2} + 3 x\right) + 3^{2}\right)\right) - 9\right) = 0$$
or
$$\left(x - 3\right) \left(x^{2} + 7 x + 12\right) = 0$$
then:
$$x_{1} = 3$$
and also
we get the equation
$$x^{2} + 7 x + 12 = 0$$
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
$$a = 1$$
$$b = 7$$
$$c = 12$$
, then
D = b^2 - 4 * a * c =
(7)^2 - 4 * (1) * (12) = 1
Because D > 0, then the equation has two roots.
x2 = (-b + sqrt(D)) / (2*a)
x3 = (-b - sqrt(D)) / (2*a)
or
$$x_{2} = -3$$
Simplify$$x_{3} = -4$$
SimplifyThe final answer for (x^3 + 4*x^2 - 9*x - 1*36) + 0 = 0:
$$x_{1} = 3$$
$$x_{2} = -3$$
$$x_{3} = -4$$