12x^2-5x-40=0

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Solution for 12x^2-5x-40=0 equation: 


Simplifying
12x2 + -5x + -40 = 0

Reorder the terms:
-40 + -5x + 12x2 = 0

Solving
-40 + -5x + 12x2 = 0

Solving for variable 'x'.

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

Divide each side by '12'.
-3.333333333 + -0.4166666667x + x2 = 0

Move the constant term to the right:

Add '3.333333333' to each side of the equation.
-3.333333333 + -0.4166666667x + 3.333333333 + x2 = 0 + 3.333333333

Reorder the terms:
-3.333333333 + 3.333333333 + -0.4166666667x + x2 = 0 + 3.333333333

Combine like terms: -3.333333333 + 3.333333333 = 0.000000000
0.000000000 + -0.4166666667x + x2 = 0 + 3.333333333
-0.4166666667x + x2 = 0 + 3.333333333

Combine like terms: 0 + 3.333333333 = 3.333333333
-0.4166666667x + x2 = 3.333333333

The x term is -0.4166666667x.  Take half its coefficient (-0.2083333334).
Square it (0.04340277781) and add it to both sides.

Add '0.04340277781' to each side of the equation.
-0.4166666667x + 0.04340277781 + x2 = 3.333333333 + 0.04340277781

Reorder the terms:
0.04340277781 + -0.4166666667x + x2 = 3.333333333 + 0.04340277781

Combine like terms: 3.333333333 + 0.04340277781 = 3.37673611081
0.04340277781 + -0.4166666667x + x2 = 3.37673611081

Factor a perfect square on the left side:
(x + -0.2083333334)(x + -0.2083333334) = 3.37673611081

Calculate the square root of the right side: 1.837589756

Break this problem into two subproblems by setting 
(x + -0.2083333334) equal to 1.837589756 and -1.837589756.


Subproblem 1
x + -0.2083333334 = 1.837589756

Simplifying
x + -0.2083333334 = 1.837589756

Reorder the terms:
-0.2083333334 + x = 1.837589756

Solving
-0.2083333334 + x = 1.837589756

Solving for variable 'x'.

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

Add '0.2083333334' to each side of the equation.
-0.2083333334 + 0.2083333334 + x = 1.837589756 + 0.2083333334

Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000
0.0000000000 + x = 1.837589756 + 0.2083333334
x = 1.837589756 + 0.2083333334

Combine like terms: 1.837589756 + 0.2083333334 = 2.0459230894
x = 2.0459230894

Simplifying
x = 2.0459230894


Subproblem 2
x + -0.2083333334 = -1.837589756

Simplifying
x + -0.2083333334 = -1.837589756

Reorder the terms:
-0.2083333334 + x = -1.837589756

Solving
-0.2083333334 + x = -1.837589756

Solving for variable 'x'.

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

Add '0.2083333334' to each side of the equation.
-0.2083333334 + 0.2083333334 + x = -1.837589756 + 0.2083333334

Combine like terms: -0.2083333334 + 0.2083333334 = 0.0000000000
0.0000000000 + x = -1.837589756 + 0.2083333334
x = -1.837589756 + 0.2083333334

Combine like terms: -1.837589756 + 0.2083333334 = -1.6292564226
x = -1.6292564226

Simplifying
x = -1.6292564226


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.0459230894, -1.6292564226}

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