x(.5x+12)=90

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Solution for x(.5x+12)=90 equation:


Simplifying
x(0.5x + 12) = 90

Reorder the terms:
x(12 + 0.5x) = 90
(12 * x + 0.5x * x) = 90
(12x + 0.5x2) = 90

Solving
12x + 0.5x2 = 90

Solving for variable 'x'.

Reorder the terms:
-90 + 12x + 0.5x2 = 90 + -90

Combine like terms: 90 + -90 = 0
-90 + 12x + 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'.
-180 + 24x + x2 = 0

Move the constant term to the right:

Add '180' to each side of the equation.
-180 + 24x + 180 + x2 = 0 + 180

Reorder the terms:
-180 + 180 + 24x + x2 = 0 + 180

Combine like terms: -180 + 180 = 0
0 + 24x + x2 = 0 + 180
24x + x2 = 0 + 180

Combine like terms: 0 + 180 = 180
24x + x2 = 180

The x term is 24x.  Take half its coefficient (12).
Square it (144) and add it to both sides.

Add '144' to each side of the equation.
24x + 144 + x2 = 180 + 144

Reorder the terms:
144 + 24x + x2 = 180 + 144

Combine like terms: 180 + 144 = 324
144 + 24x + x2 = 324

Factor a perfect square on the left side:
(x + 12)(x + 12) = 324

Calculate the square root of the right side: 18

Break this problem into two subproblems by setting 
(x + 12) equal to 18 and -18.

Subproblem 1

x + 12 = 18 Simplifying x + 12 = 18 Reorder the terms: 12 + x = 18 Solving 12 + x = 18 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = 18 + -12 Combine like terms: 12 + -12 = 0 0 + x = 18 + -12 x = 18 + -12 Combine like terms: 18 + -12 = 6 x = 6 Simplifying x = 6

Subproblem 2

x + 12 = -18 Simplifying x + 12 = -18 Reorder the terms: 12 + x = -18 Solving 12 + x = -18 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-12' to each side of the equation. 12 + -12 + x = -18 + -12 Combine like terms: 12 + -12 = 0 0 + x = -18 + -12 x = -18 + -12 Combine like terms: -18 + -12 = -30 x = -30 Simplifying x = -30

Solution

The solution to the problem is based on the solutions from the subproblems. x = {6, -30}

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