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