x^2+5x-178=0

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Solution for x^2+5x-178=0 equation: 


Simplifying
x2 + 5x + -178 = 0

Reorder the terms:
-178 + 5x + x2 = 0

Solving
-178 + 5x + x2 = 0

Solving for variable 'x'.

Begin completing the square.

Move the constant term to the right:

Add '178' to each side of the equation.
-178 + 5x + 178 + x2 = 0 + 178

Reorder the terms:
-178 + 178 + 5x + x2 = 0 + 178

Combine like terms: -178 + 178 = 0
0 + 5x + x2 = 0 + 178
5x + x2 = 0 + 178

Combine like terms: 0 + 178 = 178
5x + x2 = 178

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 = 178 + 6.25

Reorder the terms:
6.25 + 5x + x2 = 178 + 6.25

Combine like terms: 178 + 6.25 = 184.25
6.25 + 5x + x2 = 184.25

Factor a perfect square on the left side:
(x + 2.5)(x + 2.5) = 184.25

Calculate the square root of the right side: 13.57387196

Break this problem into two subproblems by setting 
(x + 2.5) equal to 13.57387196 and -13.57387196.


Subproblem 1
x + 2.5 = 13.57387196

Simplifying
x + 2.5 = 13.57387196

Reorder the terms:
2.5 + x = 13.57387196

Solving
2.5 + x = 13.57387196

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 = 13.57387196 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = 13.57387196 + -2.5
x = 13.57387196 + -2.5

Combine like terms: 13.57387196 + -2.5 = 11.07387196
x = 11.07387196

Simplifying
x = 11.07387196


Subproblem 2
x + 2.5 = -13.57387196

Simplifying
x + 2.5 = -13.57387196

Reorder the terms:
2.5 + x = -13.57387196

Solving
2.5 + x = -13.57387196

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 = -13.57387196 + -2.5

Combine like terms: 2.5 + -2.5 = 0.0
0.0 + x = -13.57387196 + -2.5
x = -13.57387196 + -2.5

Combine like terms: -13.57387196 + -2.5 = -16.07387196
x = -16.07387196

Simplifying
x = -16.07387196


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {11.07387196, -16.07387196}

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