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Simplifying 0.5(x + 6)(x + 3) = 119 Reorder the terms: 0.5(6 + x)(x + 3) = 119 Reorder the terms: 0.5(6 + x)(3 + x) = 119 Multiply (6 + x) * (3 + x) 0.5(6(3 + x) + x(3 + x)) = 119 0.5((3 * 6 + x * 6) + x(3 + x)) = 119 0.5((18 + 6x) + x(3 + x)) = 119 0.5(18 + 6x + (3 * x + x * x)) = 119 0.5(18 + 6x + (3x + x2)) = 119 Combine like terms: 6x + 3x = 9x 0.5(18 + 9x + x2) = 119 (18 * 0.5 + 9x * 0.5 + x2 * 0.5) = 119 (9 + 4.5x + 0.5x2) = 119 Solving 9 + 4.5x + 0.5x2 = 119 Solving for variable 'x'. Reorder the terms: 9 + -119 + 4.5x + 0.5x2 = 119 + -119 Combine like terms: 9 + -119 = -110 -110 + 4.5x + 0.5x2 = 119 + -119 Combine like terms: 119 + -119 = 0 -110 + 4.5x + 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'. -220 + 9x + x2 = 0 Move the constant term to the right: Add '220' to each side of the equation. -220 + 9x + 220 + x2 = 0 + 220 Reorder the terms: -220 + 220 + 9x + x2 = 0 + 220 Combine like terms: -220 + 220 = 0 0 + 9x + x2 = 0 + 220 9x + x2 = 0 + 220 Combine like terms: 0 + 220 = 220 9x + x2 = 220 The x term is 9x. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9x + 20.25 + x2 = 220 + 20.25 Reorder the terms: 20.25 + 9x + x2 = 220 + 20.25 Combine like terms: 220 + 20.25 = 240.25 20.25 + 9x + x2 = 240.25 Factor a perfect square on the left side: (x + 4.5)(x + 4.5) = 240.25 Calculate the square root of the right side: 15.5 Break this problem into two subproblems by setting (x + 4.5) equal to 15.5 and -15.5.Subproblem 1
x + 4.5 = 15.5 Simplifying x + 4.5 = 15.5 Reorder the terms: 4.5 + x = 15.5 Solving 4.5 + x = 15.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = 15.5 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = 15.5 + -4.5 x = 15.5 + -4.5 Combine like terms: 15.5 + -4.5 = 11 x = 11 Simplifying x = 11Subproblem 2
x + 4.5 = -15.5 Simplifying x + 4.5 = -15.5 Reorder the terms: 4.5 + x = -15.5 Solving 4.5 + x = -15.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + x = -15.5 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + x = -15.5 + -4.5 x = -15.5 + -4.5 Combine like terms: -15.5 + -4.5 = -20 x = -20 Simplifying x = -20Solution
The solution to the problem is based on the solutions from the subproblems. x = {11, -20}
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