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2x^2-5x-12=0 equation

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Numerical solution:

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The solution

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   2               
2*x  - 5*x - 12 = 0
$$\left(2 x^{2} - 5 x\right) - 12 = 0$$
Detail solution
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 2$$
$$b = -5$$
$$c = -12$$
, then
D = b^2 - 4 * a * c = 

(-5)^2 - 4 * (2) * (-12) = 121

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
$$x_{1} = 4$$
$$x_{2} = - \frac{3}{2}$$
Vieta's Theorem
rewrite the equation
$$\left(2 x^{2} - 5 x\right) - 12 = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$x^{2} - \frac{5 x}{2} - 6 = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - \frac{5}{2}$$
$$q = \frac{c}{a}$$
$$q = -6$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = \frac{5}{2}$$
$$x_{1} x_{2} = -6$$
The graph
Sum and product of roots [src]
sum
4 - 3/2
$$- \frac{3}{2} + 4$$
=
5/2
$$\frac{5}{2}$$
product
4*(-3)
------
  2   
$$\frac{\left(-3\right) 4}{2}$$
=
-6
$$-6$$
-6
Rapid solution [src]
x1 = -3/2
$$x_{1} = - \frac{3}{2}$$
x2 = 4
$$x_{2} = 4$$
x2 = 4
Numerical answer [src]
x1 = 4.0
x2 = -1.5
x2 = -1.5