x(.5x-1)=112

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Solution for x(.5x-1)=112 equation:


Simplifying
x(0.5x + -1) = 112

Reorder the terms:
x(-1 + 0.5x) = 112
(-1 * x + 0.5x * x) = 112
(-1x + 0.5x2) = 112

Solving
-1x + 0.5x2 = 112

Solving for variable 'x'.

Reorder the terms:
-112 + -1x + 0.5x2 = 112 + -112

Combine like terms: 112 + -112 = 0
-112 + -1x + 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'.
-224 + -2x + x2 = 0

Move the constant term to the right:

Add '224' to each side of the equation.
-224 + -2x + 224 + x2 = 0 + 224

Reorder the terms:
-224 + 224 + -2x + x2 = 0 + 224

Combine like terms: -224 + 224 = 0
0 + -2x + x2 = 0 + 224
-2x + x2 = 0 + 224

Combine like terms: 0 + 224 = 224
-2x + x2 = 224

The x term is -2x.  Take half its coefficient (-1).
Square it (1) and add it to both sides.

Add '1' to each side of the equation.
-2x + 1 + x2 = 224 + 1

Reorder the terms:
1 + -2x + x2 = 224 + 1

Combine like terms: 224 + 1 = 225
1 + -2x + x2 = 225

Factor a perfect square on the left side:
(x + -1)(x + -1) = 225

Calculate the square root of the right side: 15

Break this problem into two subproblems by setting 
(x + -1) equal to 15 and -15.

Subproblem 1

x + -1 = 15 Simplifying x + -1 = 15 Reorder the terms: -1 + x = 15 Solving -1 + x = 15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 15 + 1 Combine like terms: -1 + 1 = 0 0 + x = 15 + 1 x = 15 + 1 Combine like terms: 15 + 1 = 16 x = 16 Simplifying x = 16

Subproblem 2

x + -1 = -15 Simplifying x + -1 = -15 Reorder the terms: -1 + x = -15 Solving -1 + x = -15 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -15 + 1 Combine like terms: -1 + 1 = 0 0 + x = -15 + 1 x = -15 + 1 Combine like terms: -15 + 1 = -14 x = -14 Simplifying x = -14

Solution

The solution to the problem is based on the solutions from the subproblems. x = {16, -14}

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