5x(x+1)=75

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Solution for 5x(x+1)=75 equation: 


Simplifying
5x(x + 1) = 75

Reorder the terms:
5x(1 + x) = 75
(1 * 5x + x * 5x) = 75
(5x + 5x2) = 75

Solving
5x + 5x2 = 75

Solving for variable 'x'.

Reorder the terms:
-75 + 5x + 5x2 = 75 + -75

Combine like terms: 75 + -75 = 0
-75 + 5x + 5x2 = 0

Factor out the Greatest Common Factor (GCF), '5'.
5(-15 + x + x2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(-15 + x + x2)' equal to zero and attempt to solve:

Simplifying
-15 + x + x2 = 0

Solving
-15 + x + x2 = 0

Begin completing the square.

Move the constant term to the right:

Add '15' to each side of the equation.
-15 + x + 15 + x2 = 0 + 15

Reorder the terms:
-15 + 15 + x + x2 = 0 + 15

Combine like terms: -15 + 15 = 0
0 + x + x2 = 0 + 15
x + x2 = 0 + 15

Combine like terms: 0 + 15 = 15
x + x2 = 15

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 = 15 + 0.25

Reorder the terms:
0.25 + x + x2 = 15 + 0.25

Combine like terms: 15 + 0.25 = 15.25
0.25 + x + x2 = 15.25

Factor a perfect square on the left side:
(x + 0.5)(x + 0.5) = 15.25

Calculate the square root of the right side: 3.905124838

Break this problem into two subproblems by setting 
(x + 0.5) equal to 3.905124838 and -3.905124838.


Subproblem 1
x + 0.5 = 3.905124838

Simplifying
x + 0.5 = 3.905124838

Reorder the terms:
0.5 + x = 3.905124838

Solving
0.5 + x = 3.905124838

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 = 3.905124838 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = 3.905124838 + -0.5
x = 3.905124838 + -0.5

Combine like terms: 3.905124838 + -0.5 = 3.405124838
x = 3.405124838

Simplifying
x = 3.405124838


Subproblem 2
x + 0.5 = -3.905124838

Simplifying
x + 0.5 = -3.905124838

Reorder the terms:
0.5 + x = -3.905124838

Solving
0.5 + x = -3.905124838

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 = -3.905124838 + -0.5

Combine like terms: 0.5 + -0.5 = 0.0
0.0 + x = -3.905124838 + -0.5
x = -3.905124838 + -0.5

Combine like terms: -3.905124838 + -0.5 = -4.405124838
x = -4.405124838

Simplifying
x = -4.405124838


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {3.405124838, -4.405124838}
Solution
x = {3.405124838, -4.405124838}

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