4x^2+78x+216=0

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Solution for 4x^2+78x+216=0 equation: 


Simplifying
4x2 + 78x + 216 = 0

Reorder the terms:
216 + 78x + 4x2 = 0

Solving
216 + 78x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(108 + 39x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(108 + 39x + 2x2)' equal to zero and attempt to solve:

Simplifying
108 + 39x + 2x2 = 0

Solving
108 + 39x + 2x2 = 0

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

Divide each side by '2'.
54 + 19.5x + x2 = 0

Move the constant term to the right:

Add '-54' to each side of the equation.
54 + 19.5x + -54 + x2 = 0 + -54

Reorder the terms:
54 + -54 + 19.5x + x2 = 0 + -54

Combine like terms: 54 + -54 = 0
0 + 19.5x + x2 = 0 + -54
19.5x + x2 = 0 + -54

Combine like terms: 0 + -54 = -54
19.5x + x2 = -54

The x term is 19.5x.  Take half its coefficient (9.75).
Square it (95.0625) and add it to both sides.

Add '95.0625' to each side of the equation.
19.5x + 95.0625 + x2 = -54 + 95.0625

Reorder the terms:
95.0625 + 19.5x + x2 = -54 + 95.0625

Combine like terms: -54 + 95.0625 = 41.0625
95.0625 + 19.5x + x2 = 41.0625

Factor a perfect square on the left side:
(x + 9.75)(x + 9.75) = 41.0625

Calculate the square root of the right side: 6.408002809

Break this problem into two subproblems by setting 
(x + 9.75) equal to 6.408002809 and -6.408002809.


Subproblem 1
x + 9.75 = 6.408002809

Simplifying
x + 9.75 = 6.408002809

Reorder the terms:
9.75 + x = 6.408002809

Solving
9.75 + x = 6.408002809

Solving for variable 'x'.

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

Add '-9.75' to each side of the equation.
9.75 + -9.75 + x = 6.408002809 + -9.75

Combine like terms: 9.75 + -9.75 = 0.00
0.00 + x = 6.408002809 + -9.75
x = 6.408002809 + -9.75

Combine like terms: 6.408002809 + -9.75 = -3.341997191
x = -3.341997191

Simplifying
x = -3.341997191


Subproblem 2
x + 9.75 = -6.408002809

Simplifying
x + 9.75 = -6.408002809

Reorder the terms:
9.75 + x = -6.408002809

Solving
9.75 + x = -6.408002809

Solving for variable 'x'.

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

Add '-9.75' to each side of the equation.
9.75 + -9.75 + x = -6.408002809 + -9.75

Combine like terms: 9.75 + -9.75 = 0.00
0.00 + x = -6.408002809 + -9.75
x = -6.408002809 + -9.75

Combine like terms: -6.408002809 + -9.75 = -16.158002809
x = -16.158002809

Simplifying
x = -16.158002809


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {-3.341997191, -16.158002809}
Solution
x = {-3.341997191, -16.158002809}

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