x^2-5x-78=0

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Solution for x^2-5x-78=0 equation: 


Simplifying
x2 + -5x + -78 = 0

Reorder the terms:
-78 + -5x + x2 = 0

Solving
-78 + -5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '78' to each side of the equation.
-78 + -5x + 78 + x2 = 0 + 78

Reorder the terms:
-78 + 78 + -5x + x2 = 0 + 78

Combine like terms: -78 + 78 = 0
0 + -5x + x2 = 0 + 78
-5x + x2 = 0 + 78

Combine like terms: 0 + 78 = 78
-5x + x2 = 78

The x term is -5x.  Take half its coefficient (-2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
-5x + 6.25 + x2 = 78 + 6.25

Reorder the terms:
6.25 + -5x + x2 = 78 + 6.25

Combine like terms: 78 + 6.25 = 84.25
6.25 + -5x + x2 = 84.25

Factor a perfect square on the left side:
(x + -2.5)(x + -2.5) = 84.25

Calculate the square root of the right side: 9.178779875

Break this problem into two subproblems by setting 
(x + -2.5) equal to 9.178779875 and -9.178779875.


Subproblem 1
x + -2.5 = 9.178779875

Simplifying
x + -2.5 = 9.178779875

Reorder the terms:
-2.5 + x = 9.178779875

Solving
-2.5 + x = 9.178779875

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = 9.178779875 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = 9.178779875 + 2.5
x = 9.178779875 + 2.5

Combine like terms: 9.178779875 + 2.5 = 11.678779875
x = 11.678779875

Simplifying
x = 11.678779875


Subproblem 2
x + -2.5 = -9.178779875

Simplifying
x + -2.5 = -9.178779875

Reorder the terms:
-2.5 + x = -9.178779875

Solving
-2.5 + x = -9.178779875

Solving for variable 'x'.

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

Add '2.5' to each side of the equation.
-2.5 + 2.5 + x = -9.178779875 + 2.5

Combine like terms: -2.5 + 2.5 = 0.0
0.0 + x = -9.178779875 + 2.5
x = -9.178779875 + 2.5

Combine like terms: -9.178779875 + 2.5 = -6.678779875
x = -6.678779875

Simplifying
x = -6.678779875


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.678779875, -6.678779875}

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