4x^2-54x-126=0

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Solution for 4x^2-54x-126=0 equation: 


Simplifying
4x2 + -54x + -126 = 0

Reorder the terms:
-126 + -54x + 4x2 = 0

Solving
-126 + -54x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-63 + -27x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-63 + -27x + 2x2)' equal to zero and attempt to solve:

Simplifying
-63 + -27x + 2x2 = 0

Solving
-63 + -27x + 2x2 = 0

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

Divide each side by '2'.
-31.5 + -13.5x + x2 = 0

Move the constant term to the right:

Add '31.5' to each side of the equation.
-31.5 + -13.5x + 31.5 + x2 = 0 + 31.5

Reorder the terms:
-31.5 + 31.5 + -13.5x + x2 = 0 + 31.5

Combine like terms: -31.5 + 31.5 = 0.0
0.0 + -13.5x + x2 = 0 + 31.5
-13.5x + x2 = 0 + 31.5

Combine like terms: 0 + 31.5 = 31.5
-13.5x + x2 = 31.5

The x term is -13.5x.  Take half its coefficient (-6.75).
Square it (45.5625) and add it to both sides.

Add '45.5625' to each side of the equation.
-13.5x + 45.5625 + x2 = 31.5 + 45.5625

Reorder the terms:
45.5625 + -13.5x + x2 = 31.5 + 45.5625

Combine like terms: 31.5 + 45.5625 = 77.0625
45.5625 + -13.5x + x2 = 77.0625

Factor a perfect square on the left side:
(x + -6.75)(x + -6.75) = 77.0625

Calculate the square root of the right side: 8.778524933

Break this problem into two subproblems by setting 
(x + -6.75) equal to 8.778524933 and -8.778524933.


Subproblem 1
x + -6.75 = 8.778524933

Simplifying
x + -6.75 = 8.778524933

Reorder the terms:
-6.75 + x = 8.778524933

Solving
-6.75 + x = 8.778524933

Solving for variable 'x'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = 8.778524933 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = 8.778524933 + 6.75
x = 8.778524933 + 6.75

Combine like terms: 8.778524933 + 6.75 = 15.528524933
x = 15.528524933

Simplifying
x = 15.528524933


Subproblem 2
x + -6.75 = -8.778524933

Simplifying
x + -6.75 = -8.778524933

Reorder the terms:
-6.75 + x = -8.778524933

Solving
-6.75 + x = -8.778524933

Solving for variable 'x'.

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

Add '6.75' to each side of the equation.
-6.75 + 6.75 + x = -8.778524933 + 6.75

Combine like terms: -6.75 + 6.75 = 0.00
0.00 + x = -8.778524933 + 6.75
x = -8.778524933 + 6.75

Combine like terms: -8.778524933 + 6.75 = -2.028524933
x = -2.028524933

Simplifying
x = -2.028524933


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {15.528524933, -2.028524933}
Solution
x = {15.528524933, -2.028524933}

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