4x^2+54x+124=0

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Solution for 4x^2+54x+124=0 equation: 


Simplifying
4x2 + 54x + 124 = 0

Reorder the terms:
124 + 54x + 4x2 = 0

Solving
124 + 54x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(62 + 27x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(62 + 27x + 2x2)' equal to zero and attempt to solve:

Simplifying
62 + 27x + 2x2 = 0

Solving
62 + 27x + 2x2 = 0

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

Divide each side by '2'.
31 + 13.5x + x2 = 0

Move the constant term to the right:

Add '-31' to each side of the equation.
31 + 13.5x + -31 + x2 = 0 + -31

Reorder the terms:
31 + -31 + 13.5x + x2 = 0 + -31

Combine like terms: 31 + -31 = 0
0 + 13.5x + x2 = 0 + -31
13.5x + x2 = 0 + -31

Combine like terms: 0 + -31 = -31
13.5x + x2 = -31

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 + 45.5625

Reorder the terms:
45.5625 + 13.5x + x2 = -31 + 45.5625

Combine like terms: -31 + 45.5625 = 14.5625
45.5625 + 13.5x + x2 = 14.5625

Factor a perfect square on the left side:
(x + 6.75)(x + 6.75) = 14.5625

Calculate the square root of the right side: 3.816084381

Break this problem into two subproblems by setting 
(x + 6.75) equal to 3.816084381 and -3.816084381.


Subproblem 1
x + 6.75 = 3.816084381

Simplifying
x + 6.75 = 3.816084381

Reorder the terms:
6.75 + x = 3.816084381

Solving
6.75 + x = 3.816084381

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 = 3.816084381 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + x = 3.816084381 + -6.75
x = 3.816084381 + -6.75

Combine like terms: 3.816084381 + -6.75 = -2.933915619
x = -2.933915619

Simplifying
x = -2.933915619


Subproblem 2
x + 6.75 = -3.816084381

Simplifying
x + 6.75 = -3.816084381

Reorder the terms:
6.75 + x = -3.816084381

Solving
6.75 + x = -3.816084381

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 = -3.816084381 + -6.75

Combine like terms: 6.75 + -6.75 = 0.00
0.00 + x = -3.816084381 + -6.75
x = -3.816084381 + -6.75

Combine like terms: -3.816084381 + -6.75 = -10.566084381
x = -10.566084381

Simplifying
x = -10.566084381


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-2.933915619, -10.566084381}
Solution
x = {-2.933915619, -10.566084381}

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