4x^2+4x-78=0

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


Simplifying
4x2 + 4x + -78 = 0

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

Solving
-78 + 4x + 4x2 = 0

Solving for variable 'x'.

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

Ignore the factor 2.

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

Simplifying
-39 + 2x + 2x2 = 0

Solving
-39 + 2x + 2x2 = 0

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

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

Move the constant term to the right:

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

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

Combine like terms: -19.5 + 19.5 = 0.0
0.0 + x + x2 = 0 + 19.5
x + x2 = 0 + 19.5

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

The x term is x.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
 + 0.25 + x2 = 19.5 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 19.5 + 0.25

Combine like terms: 19.5 + 0.25 = 19.75
1.25 + x2 = 19.75

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 19.75

Calculate the square root of the right side: 4.444097209

Break this problem into two subproblems by setting 
(x + 0.5) equal to 4.444097209 and -4.444097209.


Subproblem 1
x + 0.5 = 4.444097209

Simplifying
x + 0.5 = 4.444097209

Reorder the terms:
0.5 + x = 4.444097209

Solving
0.5 + x = 4.444097209

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = 4.444097209 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 4.444097209 + -0.5
x = 4.444097209 + -0.5

Combine like terms: 4.444097209 + -0.5 = 3.944097209
x = 3.944097209

Simplifying
x = 3.944097209


Subproblem 2
x + 0.5 = -4.444097209

Simplifying
x + 0.5 = -4.444097209

Reorder the terms:
0.5 + x = -4.444097209

Solving
0.5 + x = -4.444097209

Solving for variable 'x'.

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

Add '-0.5' to each side of the equation.
0.5 + -0.5 + x = -4.444097209 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -4.444097209 + -0.5
x = -4.444097209 + -0.5

Combine like terms: -4.444097209 + -0.5 = -4.944097209
x = -4.944097209

Simplifying
x = -4.944097209


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.944097209, -4.944097209}
Solution
x = {3.944097209, -4.944097209}

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