.75x^2-5x-12=0

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Solution for .75x^2-5x-12=0 equation: 


Simplifying
0.75x2 + -5x + -12 = 0

Reorder the terms:
-12 + -5x + 0.75x2 = 0

Solving
-12 + -5x + 0.75x2 = 0

Solving for variable 'x'.

Begin completing the square.  Divide all terms by
0.75 the coefficient of the squared term: 

Divide each side by '0.75'.
-16 + -6.666666667x + x2 = 0

Move the constant term to the right:

Add '16' to each side of the equation.
-16 + -6.666666667x + 16 + x2 = 0 + 16

Reorder the terms:
-16 + 16 + -6.666666667x + x2 = 0 + 16

Combine like terms: -16 + 16 = 0
0 + -6.666666667x + x2 = 0 + 16
-6.666666667x + x2 = 0 + 16

Combine like terms: 0 + 16 = 16
-6.666666667x + x2 = 16

The x term is -6.666666667x.  Take half its coefficient (-3.333333334).
Square it (11.11111112) and add it to both sides.

Add '11.11111112' to each side of the equation.
-6.666666667x + 11.11111112 + x2 = 16 + 11.11111112

Reorder the terms:
11.11111112 + -6.666666667x + x2 = 16 + 11.11111112

Combine like terms: 16 + 11.11111112 = 27.11111112
11.11111112 + -6.666666667x + x2 = 27.11111112

Factor a perfect square on the left side:
(x + -3.333333334)(x + -3.333333334) = 27.11111112

Calculate the square root of the right side: 5.206833118

Break this problem into two subproblems by setting 
(x + -3.333333334) equal to 5.206833118 and -5.206833118.


Subproblem 1
x + -3.333333334 = 5.206833118

Simplifying
x + -3.333333334 = 5.206833118

Reorder the terms:
-3.333333334 + x = 5.206833118

Solving
-3.333333334 + x = 5.206833118

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + x = 5.206833118 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + x = 5.206833118 + 3.333333334
x = 5.206833118 + 3.333333334

Combine like terms: 5.206833118 + 3.333333334 = 8.540166452
x = 8.540166452

Simplifying
x = 8.540166452


Subproblem 2
x + -3.333333334 = -5.206833118

Simplifying
x + -3.333333334 = -5.206833118

Reorder the terms:
-3.333333334 + x = -5.206833118

Solving
-3.333333334 + x = -5.206833118

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '3.333333334' to each side of the equation.
-3.333333334 + 3.333333334 + x = -5.206833118 + 3.333333334

Combine like terms: -3.333333334 + 3.333333334 = 0.000000000
0.000000000 + x = -5.206833118 + 3.333333334
x = -5.206833118 + 3.333333334

Combine like terms: -5.206833118 + 3.333333334 = -1.873499784
x = -1.873499784

Simplifying
x = -1.873499784


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {8.540166452, -1.873499784}

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