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Simplifying 0.5(3X + 6)(X + 4) = 120 Reorder the terms: 0.5(6 + 3X)(X + 4) = 120 Reorder the terms: 0.5(6 + 3X)(4 + X) = 120 Multiply (6 + 3X) * (4 + X) 0.5(6(4 + X) + 3X * (4 + X)) = 120 0.5((4 * 6 + X * 6) + 3X * (4 + X)) = 120 0.5((24 + 6X) + 3X * (4 + X)) = 120 0.5(24 + 6X + (4 * 3X + X * 3X)) = 120 0.5(24 + 6X + (12X + 3X2)) = 120 Combine like terms: 6X + 12X = 18X 0.5(24 + 18X + 3X2) = 120 (24 * 0.5 + 18X * 0.5 + 3X2 * 0.5) = 120 (12 + 9X + 1.5X2) = 120 Solving 12 + 9X + 1.5X2 = 120 Solving for variable 'X'. Reorder the terms: 12 + -120 + 9X + 1.5X2 = 120 + -120 Combine like terms: 12 + -120 = -108 -108 + 9X + 1.5X2 = 120 + -120 Combine like terms: 120 + -120 = 0 -108 + 9X + 1.5X2 = 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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