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x^3+4x^2-9x-36=0

x^3+4x^2-9x-36=0 equation

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Numerical solution:

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The solution

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 3      2               
x  + 4*x  - 9*x - 36 = 0
$$x^{3} + 4 x^{2} - 9 x - 36 = 0$$
Detail solution
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
$$x_{3} = -4$$
Simplify
The final answer for (x^3 + 4*x^2 - 9*x - 1*36) + 0 = 0:
$$x_{1} = 3$$
$$x_{2} = -3$$
$$x_{3} = -4$$
Vieta's Theorem
it is reduced cubic equation
$$p x^{2} + q x + v + x^{3} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 4$$
$$q = \frac{c}{a}$$
$$q = -9$$
$$v = \frac{d}{a}$$
$$v = -36$$
Vieta Formulas
$$x_{1} + x_{2} + x_{3} = - p$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = q$$
$$x_{1} x_{2} x_{3} = v$$
$$x_{1} + x_{2} + x_{3} = -4$$
$$x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3} = -9$$
$$x_{1} x_{2} x_{3} = -36$$
The graph
Rapid solution [src]
x1 = -4
$$x_{1} = -4$$
x2 = -3
$$x_{2} = -3$$
x3 = 3
$$x_{3} = 3$$
Sum and product of roots [src]
sum
0 - 4 - 3 + 3
$$\left(\left(-4 + 0\right) - 3\right) + 3$$
=
-4
$$-4$$
product
1*-4*-3*3
$$1 \left(-4\right) \left(-3\right) 3$$
=
36
$$36$$
36
Numerical answer [src]
x1 = 3.0
x2 = -3.0
x3 = -4.0
x3 = -4.0
The graph
x^3+4x^2-9x-36=0 equation