12x^2+5x-4=0

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Solution for 12x^2+5x-4=0 equation:


Simplifying
12x2 + 5x + -4 = 0

Reorder the terms:
-4 + 5x + 12x2 = 0

Solving
-4 + 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'.
-0.3333333333 + 0.4166666667x + x2 = 0

Move the constant term to the right:

Add '0.3333333333' to each side of the equation.
-0.3333333333 + 0.4166666667x + 0.3333333333 + x2 = 0 + 0.3333333333

Reorder the terms:
-0.3333333333 + 0.3333333333 + 0.4166666667x + x2 = 0 + 0.3333333333

Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000
0.0000000000 + 0.4166666667x + x2 = 0 + 0.3333333333
0.4166666667x + x2 = 0 + 0.3333333333

Combine like terms: 0 + 0.3333333333 = 0.3333333333
0.4166666667x + x2 = 0.3333333333

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 = 0.3333333333 + 0.04340277781

Reorder the terms:
0.04340277781 + 0.4166666667x + x2 = 0.3333333333 + 0.04340277781

Combine like terms: 0.3333333333 + 0.04340277781 = 0.37673611111
0.04340277781 + 0.4166666667x + x2 = 0.37673611111

Factor a perfect square on the left side:
(x + 0.2083333334)(x + 0.2083333334) = 0.37673611111

Calculate the square root of the right side: 0.613788328

Break this problem into two subproblems by setting 
(x + 0.2083333334) equal to 0.613788328 and -0.613788328.


Subproblem 1
x + 0.2083333334 = 0.613788328

Simplifying
x + 0.2083333334 = 0.613788328

Reorder the terms:
0.2083333334 + x = 0.613788328

Solving
0.2083333334 + x = 0.613788328

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 = 0.613788328 + -0.2083333334

Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000
0.0000000000 + x = 0.613788328 + -0.2083333334
x = 0.613788328 + -0.2083333334

Combine like terms: 0.613788328 + -0.2083333334 = 0.4054549946
x = 0.4054549946

Simplifying
x = 0.4054549946


Subproblem 2
x + 0.2083333334 = -0.613788328

Simplifying
x + 0.2083333334 = -0.613788328

Reorder the terms:
0.2083333334 + x = -0.613788328

Solving
0.2083333334 + x = -0.613788328

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 = -0.613788328 + -0.2083333334

Combine like terms: 0.2083333334 + -0.2083333334 = 0.0000000000
0.0000000000 + x = -0.613788328 + -0.2083333334
x = -0.613788328 + -0.2083333334

Combine like terms: -0.613788328 + -0.2083333334 = -0.8221216614
x = -0.8221216614

Simplifying
x = -0.8221216614


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.4054549946, -0.8221216614}

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