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Simplifying 120 = 0.5(x + 4)(3x + 6) Reorder the terms: 120 = 0.5(4 + x)(3x + 6) Reorder the terms: 120 = 0.5(4 + x)(6 + 3x) Multiply (4 + x) * (6 + 3x) 120 = 0.5(4(6 + 3x) + x(6 + 3x)) 120 = 0.5((6 * 4 + 3x * 4) + x(6 + 3x)) 120 = 0.5((24 + 12x) + x(6 + 3x)) 120 = 0.5(24 + 12x + (6 * x + 3x * x)) 120 = 0.5(24 + 12x + (6x + 3x2)) Combine like terms: 12x + 6x = 18x 120 = 0.5(24 + 18x + 3x2) 120 = (24 * 0.5 + 18x * 0.5 + 3x2 * 0.5) 120 = (12 + 9x + 1.5x2) Solving 120 = 12 + 9x + 1.5x2 Solving for variable 'x'. Combine like terms: 120 + -12 = 108 108 + -9x + -1.5x2 = 12 + 9x + 1.5x2 + -12 + -9x + -1.5x2 Reorder the terms: 108 + -9x + -1.5x2 = 12 + -12 + 9x + -9x + 1.5x2 + -1.5x2 Combine like terms: 12 + -12 = 0 108 + -9x + -1.5x2 = 0 + 9x + -9x + 1.5x2 + -1.5x2 108 + -9x + -1.5x2 = 9x + -9x + 1.5x2 + -1.5x2 Combine like terms: 9x + -9x = 0 108 + -9x + -1.5x2 = 0 + 1.5x2 + -1.5x2 108 + -9x + -1.5x2 = 1.5x2 + -1.5x2 Combine like terms: 1.5x2 + -1.5x2 = 0.0 108 + -9x + -1.5x2 = 0.0 Begin completing the square. Divide all terms by -1.5 the coefficient of the squared term: Divide each side by '-1.5'. -72 + 6x + x2 = 0 Move the constant term to the right: Add '72' to each side of the equation. -72 + 6x + 72 + x2 = 0 + 72 Reorder the terms: -72 + 72 + 6x + x2 = 0 + 72 Combine like terms: -72 + 72 = 0 0 + 6x + x2 = 0 + 72 6x + x2 = 0 + 72 Combine like terms: 0 + 72 = 72 6x + x2 = 72 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 = 72 + 9 Reorder the terms: 9 + 6x + x2 = 72 + 9 Combine like terms: 72 + 9 = 81 9 + 6x + x2 = 81 Factor a perfect square on the left side: (x + 3)(x + 3) = 81 Calculate the square root of the right side: 9 Break this problem into two subproblems by setting (x + 3) equal to 9 and -9.Subproblem 1
x + 3 = 9 Simplifying x + 3 = 9 Reorder the terms: 3 + x = 9 Solving 3 + x = 9 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 = 9 + -3 Combine like terms: 3 + -3 = 0 0 + x = 9 + -3 x = 9 + -3 Combine like terms: 9 + -3 = 6 x = 6 Simplifying x = 6Subproblem 2
x + 3 = -9 Simplifying x + 3 = -9 Reorder the terms: 3 + x = -9 Solving 3 + x = -9 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 = -9 + -3 Combine like terms: 3 + -3 = 0 0 + x = -9 + -3 x = -9 + -3 Combine like terms: -9 + -3 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {6, -12}
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