x(x+1)=8

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Solution for x(x+1)=8 equation: 


Simplifying
x(x + 1) = 8

Reorder the terms:
x(1 + x) = 8
(1 * x + x * x) = 8
(1x + x2) = 8

Solving
1x + x2 = 8

Solving for variable 'x'.

Reorder the terms:
-8 + 1x + x2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 1x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 1x + 8 + x2 = 0 + 8

Reorder the terms:
-8 + 8 + 1x + x2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 1x + x2 = 0 + 8
1x + x2 = 0 + 8

Combine like terms: 0 + 8 = 8
1x + x2 = 8

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 = 8 + 0.25

Reorder the terms:
0.25 + x + x2 = 8 + 0.25

Combine like terms: 8 + 0.25 = 8.25
0.25 + x + x2 = 8.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 8.25

Calculate the square root of the right side: 2.872281323

Break this problem into two subproblems by setting 
(x + 0.5) equal to 2.872281323 and -2.872281323.


Subproblem 1
x + 0.5 = 2.872281323

Simplifying
x + 0.5 = 2.872281323

Reorder the terms:
0.5 + x = 2.872281323

Solving
0.5 + x = 2.872281323

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 = 2.872281323 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 2.872281323 + -0.5
x = 2.872281323 + -0.5

Combine like terms: 2.872281323 + -0.5 = 2.372281323
x = 2.372281323

Simplifying
x = 2.372281323


Subproblem 2
x + 0.5 = -2.872281323

Simplifying
x + 0.5 = -2.872281323

Reorder the terms:
0.5 + x = -2.872281323

Solving
0.5 + x = -2.872281323

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 = -2.872281323 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -2.872281323 + -0.5
x = -2.872281323 + -0.5

Combine like terms: -2.872281323 + -0.5 = -3.372281323
x = -3.372281323

Simplifying
x = -3.372281323


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.372281323, -3.372281323}

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