x(2x+5)+(2x+5-12)=108

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Solution for x(2x+5)+(2x+5-12)=108 equation:


Simplifying
x(2x + 5) + (2x + 5 + -12) = 108

Reorder the terms:
x(5 + 2x) + (2x + 5 + -12) = 108
(5 * x + 2x * x) + (2x + 5 + -12) = 108
(5x + 2x2) + (2x + 5 + -12) = 108

Reorder the terms:
5x + 2x2 + (5 + -12 + 2x) = 108

Combine like terms: 5 + -12 = -7
5x + 2x2 + (-7 + 2x) = 108

Remove parenthesis around (-7 + 2x)
5x + 2x2 + -7 + 2x = 108

Reorder the terms:
-7 + 5x + 2x + 2x2 = 108

Combine like terms: 5x + 2x = 7x
-7 + 7x + 2x2 = 108

Solving
-7 + 7x + 2x2 = 108

Solving for variable 'x'.

Reorder the terms:
-7 + -108 + 7x + 2x2 = 108 + -108

Combine like terms: -7 + -108 = -115
-115 + 7x + 2x2 = 108 + -108

Combine like terms: 108 + -108 = 0
-115 + 7x + 2x2 = 0

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

Divide each side by '2'.
-57.5 + 3.5x + x2 = 0

Move the constant term to the right:

Add '57.5' to each side of the equation.
-57.5 + 3.5x + 57.5 + x2 = 0 + 57.5

Reorder the terms:
-57.5 + 57.5 + 3.5x + x2 = 0 + 57.5

Combine like terms: -57.5 + 57.5 = 0.0
0.0 + 3.5x + x2 = 0 + 57.5
3.5x + x2 = 0 + 57.5

Combine like terms: 0 + 57.5 = 57.5
3.5x + x2 = 57.5

The x term is 3.5x.  Take half its coefficient (1.75).
Square it (3.0625) and add it to both sides.

Add '3.0625' to each side of the equation.
3.5x + 3.0625 + x2 = 57.5 + 3.0625

Reorder the terms:
3.0625 + 3.5x + x2 = 57.5 + 3.0625

Combine like terms: 57.5 + 3.0625 = 60.5625
3.0625 + 3.5x + x2 = 60.5625

Factor a perfect square on the left side:
(x + 1.75)(x + 1.75) = 60.5625

Calculate the square root of the right side: 7.782191208

Break this problem into two subproblems by setting 
(x + 1.75) equal to 7.782191208 and -7.782191208.

Subproblem 1

x + 1.75 = 7.782191208 Simplifying x + 1.75 = 7.782191208 Reorder the terms: 1.75 + x = 7.782191208 Solving 1.75 + x = 7.782191208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 7.782191208 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 7.782191208 + -1.75 x = 7.782191208 + -1.75 Combine like terms: 7.782191208 + -1.75 = 6.032191208 x = 6.032191208 Simplifying x = 6.032191208

Subproblem 2

x + 1.75 = -7.782191208 Simplifying x + 1.75 = -7.782191208 Reorder the terms: 1.75 + x = -7.782191208 Solving 1.75 + x = -7.782191208 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -7.782191208 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -7.782191208 + -1.75 x = -7.782191208 + -1.75 Combine like terms: -7.782191208 + -1.75 = -9.532191208 x = -9.532191208 Simplifying x = -9.532191208

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6.032191208, -9.532191208}

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