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Simplifying 105 = 0.5(x + 2)(3x + 15) Reorder the terms: 105 = 0.5(2 + x)(3x + 15) Reorder the terms: 105 = 0.5(2 + x)(15 + 3x) Multiply (2 + x) * (15 + 3x) 105 = 0.5(2(15 + 3x) + x(15 + 3x)) 105 = 0.5((15 * 2 + 3x * 2) + x(15 + 3x)) 105 = 0.5((30 + 6x) + x(15 + 3x)) 105 = 0.5(30 + 6x + (15 * x + 3x * x)) 105 = 0.5(30 + 6x + (15x + 3x2)) Combine like terms: 6x + 15x = 21x 105 = 0.5(30 + 21x + 3x2) 105 = (30 * 0.5 + 21x * 0.5 + 3x2 * 0.5) 105 = (15 + 10.5x + 1.5x2) Solving 105 = 15 + 10.5x + 1.5x2 Solving for variable 'x'. Combine like terms: 105 + -15 = 90 90 + -10.5x + -1.5x2 = 15 + 10.5x + 1.5x2 + -15 + -10.5x + -1.5x2 Reorder the terms: 90 + -10.5x + -1.5x2 = 15 + -15 + 10.5x + -10.5x + 1.5x2 + -1.5x2 Combine like terms: 15 + -15 = 0 90 + -10.5x + -1.5x2 = 0 + 10.5x + -10.5x + 1.5x2 + -1.5x2 90 + -10.5x + -1.5x2 = 10.5x + -10.5x + 1.5x2 + -1.5x2 Combine like terms: 10.5x + -10.5x = 0.0 90 + -10.5x + -1.5x2 = 0.0 + 1.5x2 + -1.5x2 90 + -10.5x + -1.5x2 = 1.5x2 + -1.5x2 Combine like terms: 1.5x2 + -1.5x2 = 0.0 90 + -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'. -60 + 7x + x2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 7x + 60 + x2 = 0 + 60 Reorder the terms: -60 + 60 + 7x + x2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 7x + x2 = 0 + 60 7x + x2 = 0 + 60 Combine like terms: 0 + 60 = 60 7x + x2 = 60 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 = 60 + 12.25 Reorder the terms: 12.25 + 7x + x2 = 60 + 12.25 Combine like terms: 60 + 12.25 = 72.25 12.25 + 7x + x2 = 72.25 Factor a perfect square on the left side: (x + 3.5)(x + 3.5) = 72.25 Calculate the square root of the right side: 8.5 Break this problem into two subproblems by setting (x + 3.5) equal to 8.5 and -8.5.Subproblem 1
x + 3.5 = 8.5 Simplifying x + 3.5 = 8.5 Reorder the terms: 3.5 + x = 8.5 Solving 3.5 + x = 8.5 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 = 8.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = 8.5 + -3.5 x = 8.5 + -3.5 Combine like terms: 8.5 + -3.5 = 5 x = 5 Simplifying x = 5Subproblem 2
x + 3.5 = -8.5 Simplifying x + 3.5 = -8.5 Reorder the terms: 3.5 + x = -8.5 Solving 3.5 + x = -8.5 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 = -8.5 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + x = -8.5 + -3.5 x = -8.5 + -3.5 Combine like terms: -8.5 + -3.5 = -12 x = -12 Simplifying x = -12Solution
The solution to the problem is based on the solutions from the subproblems. x = {5, -12}
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