0.5(x+6)(x)=108

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Solution for 0.5(x+6)(x)=108 equation:


Simplifying
0.5(x + 6)(x) = 108

Reorder the terms:
0.5(6 + x)(x) = 108

Reorder the terms for easier multiplication:
0.5x(6 + x) = 108
(6 * 0.5x + x * 0.5x) = 108
(3x + 0.5x2) = 108

Solving
3x + 0.5x2 = 108

Solving for variable 'x'.

Reorder the terms:
-108 + 3x + 0.5x2 = 108 + -108

Combine like terms: 108 + -108 = 0
-108 + 3x + 0.5x2 = 0

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

Divide each side by '0.5'.
-216 + 6x + x2 = 0

Move the constant term to the right:

Add '216' to each side of the equation.
-216 + 6x + 216 + x2 = 0 + 216

Reorder the terms:
-216 + 216 + 6x + x2 = 0 + 216

Combine like terms: -216 + 216 = 0
0 + 6x + x2 = 0 + 216
6x + x2 = 0 + 216

Combine like terms: 0 + 216 = 216
6x + x2 = 216

The x term is 6x.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6x + 9 + x2 = 216 + 9

Reorder the terms:
9 + 6x + x2 = 216 + 9

Combine like terms: 216 + 9 = 225
9 + 6x + x2 = 225

Factor a perfect square on the left side:
(x + 3)(x + 3) = 225

Calculate the square root of the right side: 15

Break this problem into two subproblems by setting 
(x + 3) equal to 15 and -15.

Subproblem 1

x + 3 = 15 Simplifying x + 3 = 15 Reorder the terms: 3 + x = 15 Solving 3 + x = 15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = 15 + -3 Combine like terms: 3 + -3 = 0 0 + x = 15 + -3 x = 15 + -3 Combine like terms: 15 + -3 = 12 x = 12 Simplifying x = 12

Subproblem 2

x + 3 = -15 Simplifying x + 3 = -15 Reorder the terms: 3 + x = -15 Solving 3 + x = -15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + x = -15 + -3 Combine like terms: 3 + -3 = 0 0 + x = -15 + -3 x = -15 + -3 Combine like terms: -15 + -3 = -18 x = -18 Simplifying x = -18

Solution

The solution to the problem is based on the solutions from the subproblems. x = {12, -18}

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