4x(x+1)=40

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Solution for 4x(x+1)=40 equation: 


Simplifying
4x(x + 1) = 40

Reorder the terms:
4x(1 + x) = 40
(1 * 4x + x * 4x) = 40
(4x + 4x2) = 40

Solving
4x + 4x2 = 40

Solving for variable 'x'.

Reorder the terms:
-40 + 4x + 4x2 = 40 + -40

Combine like terms: 40 + -40 = 0
-40 + 4x + 4x2 = 0

Factor out the Greatest Common Factor (GCF), '4'.
4(-10 + x + x2) = 0

Ignore the factor 4.

Subproblem 1
Set the factor '(-10 + x + x2)' equal to zero and attempt to solve:

Simplifying
-10 + x + x2 = 0

Solving
-10 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '10' to each side of the equation.
-10 + x + 10 + x2 = 0 + 10

Reorder the terms:
-10 + 10 + x + x2 = 0 + 10

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

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

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.
x + 0.25 + x2 = 10 + 0.25

Reorder the terms:
0.25 + x + x2 = 10 + 0.25

Combine like terms: 10 + 0.25 = 10.25
0.25 + x + x2 = 10.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 10.25

Calculate the square root of the right side: 3.201562119

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.201562119 and -3.201562119.


Subproblem 1
x + 0.5 = 3.201562119

Simplifying
x + 0.5 = 3.201562119

Reorder the terms:
0.5 + x = 3.201562119

Solving
0.5 + x = 3.201562119

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 = 3.201562119 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.201562119 + -0.5
x = 3.201562119 + -0.5

Combine like terms: 3.201562119 + -0.5 = 2.701562119
x = 2.701562119

Simplifying
x = 2.701562119


Subproblem 2
x + 0.5 = -3.201562119

Simplifying
x + 0.5 = -3.201562119

Reorder the terms:
0.5 + x = -3.201562119

Solving
0.5 + x = -3.201562119

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 = -3.201562119 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.201562119 + -0.5
x = -3.201562119 + -0.5

Combine like terms: -3.201562119 + -0.5 = -3.701562119
x = -3.701562119

Simplifying
x = -3.701562119


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {2.701562119, -3.701562119}
Solution
x = {2.701562119, -3.701562119}

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