x(x+1)=176

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Solution for x(x+1)=176 equation: 


Simplifying
x(x + 1) = 176

Reorder the terms:
x(1 + x) = 176
(1 * x + x * x) = 176
(1x + x2) = 176

Solving
1x + x2 = 176

Solving for variable 'x'.

Reorder the terms:
-176 + 1x + x2 = 176 + -176

Combine like terms: 176 + -176 = 0
-176 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '176' to each side of the equation.
-176 + 1x + 176 + x2 = 0 + 176

Reorder the terms:
-176 + 176 + 1x + x2 = 0 + 176

Combine like terms: -176 + 176 = 0
0 + 1x + x2 = 0 + 176
1x + x2 = 0 + 176

Combine like terms: 0 + 176 = 176
1x + x2 = 176

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.
x + 0.25 + x2 = 176 + 0.25

Reorder the terms:
0.25 + x + x2 = 176 + 0.25

Combine like terms: 176 + 0.25 = 176.25
0.25 + x + x2 = 176.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 176.25

Calculate the square root of the right side: 13.275918047

Break this problem into two subproblems by setting 
(x + 0.5) equal to 13.275918047 and -13.275918047.


Subproblem 1
x + 0.5 = 13.275918047

Simplifying
x + 0.5 = 13.275918047

Reorder the terms:
0.5 + x = 13.275918047

Solving
0.5 + x = 13.275918047

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 = 13.275918047 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 13.275918047 + -0.5
x = 13.275918047 + -0.5

Combine like terms: 13.275918047 + -0.5 = 12.775918047
x = 12.775918047

Simplifying
x = 12.775918047


Subproblem 2
x + 0.5 = -13.275918047

Simplifying
x + 0.5 = -13.275918047

Reorder the terms:
0.5 + x = -13.275918047

Solving
0.5 + x = -13.275918047

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 = -13.275918047 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -13.275918047 + -0.5
x = -13.275918047 + -0.5

Combine like terms: -13.275918047 + -0.5 = -13.775918047
x = -13.775918047

Simplifying
x = -13.775918047


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {12.775918047, -13.775918047}

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