12x^2+5x-33=0

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Solution for 12x^2+5x-33=0 equation: 


Simplifying
12x2 + 5x + -33 = 0

Reorder the terms:
-33 + 5x + 12x2 = 0

Solving
-33 + 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'.
-2.75 + 0.4166666667x + x2 = 0

Move the constant term to the right:

Add '2.75' to each side of the equation.
-2.75 + 0.4166666667x + 2.75 + x2 = 0 + 2.75

Reorder the terms:
-2.75 + 2.75 + 0.4166666667x + x2 = 0 + 2.75

Combine like terms: -2.75 + 2.75 = 0.00
0.00 + 0.4166666667x + x2 = 0 + 2.75
0.4166666667x + x2 = 0 + 2.75

Combine like terms: 0 + 2.75 = 2.75
0.4166666667x + x2 = 2.75

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 = 2.75 + 0.04340277781

Reorder the terms:
0.04340277781 + 0.4166666667x + x2 = 2.75 + 0.04340277781

Combine like terms: 2.75 + 0.04340277781 = 2.79340277781
0.04340277781 + 0.4166666667x + x2 = 2.79340277781

Factor a perfect square on the left side:
(x + 0.2083333334)(x + 0.2083333334) = 2.79340277781

Calculate the square root of the right side: 1.671347593

Break this problem into two subproblems by setting 
(x + 0.2083333334) equal to 1.671347593 and -1.671347593.


Subproblem 1
x + 0.2083333334 = 1.671347593

Simplifying
x + 0.2083333334 = 1.671347593

Reorder the terms:
0.2083333334 + x = 1.671347593

Solving
0.2083333334 + x = 1.671347593

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 = 1.671347593 + -0.2083333334

Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000
0.0000000000 + x = 1.671347593 + -0.2083333334
x = 1.671347593 + -0.2083333334

Combine like terms: 1.671347593 + -0.2083333334 = 1.4630142596
x = 1.4630142596

Simplifying
x = 1.4630142596


Subproblem 2
x + 0.2083333334 = -1.671347593

Simplifying
x + 0.2083333334 = -1.671347593

Reorder the terms:
0.2083333334 + x = -1.671347593

Solving
0.2083333334 + x = -1.671347593

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 = -1.671347593 + -0.2083333334

Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000
0.0000000000 + x = -1.671347593 + -0.2083333334
x = -1.671347593 + -0.2083333334

Combine like terms: -1.671347593 + -0.2083333334 = -1.8796809264
x = -1.8796809264

Simplifying
x = -1.8796809264


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {1.4630142596, -1.8796809264}

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