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