2x^2-2x-78=0

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


Simplifying
2x2 + -2x + -78 = 0

Reorder the terms:
-78 + -2x + 2x2 = 0

Solving
-78 + -2x + 2x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-39 + -1x + x2) = 0

Ignore the factor 2.

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

Simplifying
-39 + -1x + x2 = 0

Solving
-39 + -1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + -1x + 39 + x2 = 0 + 39

Reorder the terms:
-39 + 39 + -1x + x2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + -1x + x2 = 0 + 39
-1x + x2 = 0 + 39

Combine like terms: 0 + 39 = 39
-1x + x2 = 39

The x term is -1x.  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.
-1x + 0.25 + x2 = 39 + 0.25

Reorder the terms:
0.25 + -1x + x2 = 39 + 0.25

Combine like terms: 39 + 0.25 = 39.25
0.25 + -1x + x2 = 39.25

Factor a perfect square on the left side:
(x + -0.5)(x + -0.5) = 39.25

Calculate the square root of the right side: 6.264982043

Break this problem into two subproblems by setting 
(x + -0.5) equal to 6.264982043 and -6.264982043.


Subproblem 1
x + -0.5 = 6.264982043

Simplifying
x + -0.5 = 6.264982043

Reorder the terms:
-0.5 + x = 6.264982043

Solving
-0.5 + x = 6.264982043

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 = 6.264982043 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = 6.264982043 + 0.5
x = 6.264982043 + 0.5

Combine like terms: 6.264982043 + 0.5 = 6.764982043
x = 6.764982043

Simplifying
x = 6.764982043


Subproblem 2
x + -0.5 = -6.264982043

Simplifying
x + -0.5 = -6.264982043

Reorder the terms:
-0.5 + x = -6.264982043

Solving
-0.5 + x = -6.264982043

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 = -6.264982043 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + x = -6.264982043 + 0.5
x = -6.264982043 + 0.5

Combine like terms: -6.264982043 + 0.5 = -5.764982043
x = -5.764982043

Simplifying
x = -5.764982043


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {6.764982043, -5.764982043}
Solution
x = {6.764982043, -5.764982043}

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