10x(x+1)=8

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Solution for 10x(x+1)=8 equation: 


Simplifying
10x(x + 1) = 8

Reorder the terms:
10x(1 + x) = 8
(1 * 10x + x * 10x) = 8
(10x + 10x2) = 8

Solving
10x + 10x2 = 8

Solving for variable 'x'.

Reorder the terms:
-8 + 10x + 10x2 = 8 + -8

Combine like terms: 8 + -8 = 0
-8 + 10x + 10x2 = 0

Factor out the Greatest Common Factor (GCF), '2'.
2(-4 + 5x + 5x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-4 + 5x + 5x2)' equal to zero and attempt to solve:

Simplifying
-4 + 5x + 5x2 = 0

Solving
-4 + 5x + 5x2 = 0

Begin completing the square.  Divide all terms by
5 the coefficient of the squared term: 

Divide each side by '5'.
-0.8 + x + x2 = 0

Move the constant term to the right:

Add '0.8' to each side of the equation.
-0.8 + x + 0.8 + x2 = 0 + 0.8

Reorder the terms:
-0.8 + 0.8 + x + x2 = 0 + 0.8

Combine like terms: -0.8 + 0.8 = 0.0
0.0 + x + x2 = 0 + 0.8
x + x2 = 0 + 0.8

Combine like terms: 0 + 0.8 = 0.8
x + x2 = 0.8

The x term is x.  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.
 + 0.25 + x2 = 0.8 + 0.25

Combine like terms:  + 0.25 = 1.25
1.25 + x2 = 0.8 + 0.25

Combine like terms: 0.8 + 0.25 = 1.05
1.25 + x2 = 1.05

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 1.05

Calculate the square root of the right side: 1.024695077

Break this problem into two subproblems by setting 
(x + 0.5) equal to 1.024695077 and -1.024695077.


Subproblem 1
x + 0.5 = 1.024695077

Simplifying
x + 0.5 = 1.024695077

Reorder the terms:
0.5 + x = 1.024695077

Solving
0.5 + x = 1.024695077

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 = 1.024695077 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 1.024695077 + -0.5
x = 1.024695077 + -0.5

Combine like terms: 1.024695077 + -0.5 = 0.524695077
x = 0.524695077

Simplifying
x = 0.524695077


Subproblem 2
x + 0.5 = -1.024695077

Simplifying
x + 0.5 = -1.024695077

Reorder the terms:
0.5 + x = -1.024695077

Solving
0.5 + x = -1.024695077

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 = -1.024695077 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -1.024695077 + -0.5
x = -1.024695077 + -0.5

Combine like terms: -1.024695077 + -0.5 = -1.524695077
x = -1.524695077

Simplifying
x = -1.524695077


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {0.524695077, -1.524695077}
Solution
x = {0.524695077, -1.524695077}

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