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Simplifying 60 = 0.5(x + 5)(3x + 6) Reorder the terms: 60 = 0.5(5 + x)(3x + 6) Reorder the terms: 60 = 0.5(5 + x)(6 + 3x) Multiply (5 + x) * (6 + 3x) 60 = 0.5(5(6 + 3x) + x(6 + 3x)) 60 = 0.5((6 * 5 + 3x * 5) + x(6 + 3x)) 60 = 0.5((30 + 15x) + x(6 + 3x)) 60 = 0.5(30 + 15x + (6 * x + 3x * x)) 60 = 0.5(30 + 15x + (6x + 3x2)) Combine like terms: 15x + 6x = 21x 60 = 0.5(30 + 21x + 3x2) 60 = (30 * 0.5 + 21x * 0.5 + 3x2 * 0.5) 60 = (15 + 10.5x + 1.5x2) Solving 60 = 15 + 10.5x + 1.5x2 Solving for variable 'x'. Combine like terms: 60 + -15 = 45 45 + -10.5x + -1.5x2 = 15 + 10.5x + 1.5x2 + -15 + -10.5x + -1.5x2 Reorder the terms: 45 + -10.5x + -1.5x2 = 15 + -15 + 10.5x + -10.5x + 1.5x2 + -1.5x2 Combine like terms: 15 + -15 = 0 45 + -10.5x + -1.5x2 = 0 + 10.5x + -10.5x + 1.5x2 + -1.5x2 45 + -10.5x + -1.5x2 = 10.5x + -10.5x + 1.5x2 + -1.5x2 Combine like terms: 10.5x + -10.5x = 0.0 45 + -10.5x + -1.5x2 = 0.0 + 1.5x2 + -1.5x2 45 + -10.5x + -1.5x2 = 1.5x2 + -1.5x2 Combine like terms: 1.5x2 + -1.5x2 = 0.0 45 + -10.5x + -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'. -30 + 7x + x2 = 0 Move the constant term to the right: Add '30' to each side of the equation. -30 + 7x + 30 + x2 = 0 + 30 Reorder the terms: -30 + 30 + 7x + x2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 7x + x2 = 0 + 30 7x + x2 = 0 + 30 Combine like terms: 0 + 30 = 30 7x + x2 = 30 The x term is 7x. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7x + 12.25 + x2 = 30 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 30 + 12.25 Combine like terms: 30 + 12.25 = 42.25 12.25 + 7x + x2 = 42.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 42.25 Calculate the square root of the right side: 6.5 Break this problem into two subproblems by setting (x + 3.5) equal to 6.5 and -6.5.Subproblem 1
x + 3.5 = 6.5 Simplifying x + 3.5 = 6.5 Reorder the terms: 3.5 + x = 6.5 Solving 3.5 + x = 6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = 6.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 6.5 + -3.5 x = 6.5 + -3.5 Combine like terms: 6.5 + -3.5 = 3 x = 3 Simplifying x = 3Subproblem 2
x + 3.5 = -6.5 Simplifying x + 3.5 = -6.5 Reorder the terms: 3.5 + x = -6.5 Solving 3.5 + x = -6.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + x = -6.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -6.5 + -3.5 x = -6.5 + -3.5 Combine like terms: -6.5 + -3.5 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {3, -10}
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