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Simplifying (2x + 12)(2x + 15) = 500 Reorder the terms: (12 + 2x)(2x + 15) = 500 Reorder the terms: (12 + 2x)(15 + 2x) = 500 Multiply (12 + 2x) * (15 + 2x) (12(15 + 2x) + 2x * (15 + 2x)) = 500 ((15 * 12 + 2x * 12) + 2x * (15 + 2x)) = 500 ((180 + 24x) + 2x * (15 + 2x)) = 500 (180 + 24x + (15 * 2x + 2x * 2x)) = 500 (180 + 24x + (30x + 4x2)) = 500 Combine like terms: 24x + 30x = 54x (180 + 54x + 4x2) = 500 Solving 180 + 54x + 4x2 = 500 Solving for variable 'x'. Reorder the terms: 180 + -500 + 54x + 4x2 = 500 + -500 Combine like terms: 180 + -500 = -320 -320 + 54x + 4x2 = 500 + -500 Combine like terms: 500 + -500 = 0 -320 + 54x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-160 + 27x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-160 + 27x + 2x2)' equal to zero and attempt to solve: Simplifying -160 + 27x + 2x2 = 0 Solving -160 + 27x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -80 + 13.5x + x2 = 0 Move the constant term to the right: Add '80' to each side of the equation. -80 + 13.5x + 80 + x2 = 0 + 80 Reorder the terms: -80 + 80 + 13.5x + x2 = 0 + 80 Combine like terms: -80 + 80 = 0 0 + 13.5x + x2 = 0 + 80 13.5x + x2 = 0 + 80 Combine like terms: 0 + 80 = 80 13.5x + x2 = 80 The x term is 13.5x. Take half its coefficient (6.75). Square it (45.5625) and add it to both sides. Add '45.5625' to each side of the equation. 13.5x + 45.5625 + x2 = 80 + 45.5625 Reorder the terms: 45.5625 + 13.5x + x2 = 80 + 45.5625 Combine like terms: 80 + 45.5625 = 125.5625 45.5625 + 13.5x + x2 = 125.5625 Factor a perfect square on the left side: (x + 6.75)(x + 6.75) = 125.5625 Calculate the square root of the right side: 11.205467416 Break this problem into two subproblems by setting (x + 6.75) equal to 11.205467416 and -11.205467416.Subproblem 1
x + 6.75 = 11.205467416 Simplifying x + 6.75 = 11.205467416 Reorder the terms: 6.75 + x = 11.205467416 Solving 6.75 + x = 11.205467416 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + x = 11.205467416 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + x = 11.205467416 + -6.75 x = 11.205467416 + -6.75 Combine like terms: 11.205467416 + -6.75 = 4.455467416 x = 4.455467416 Simplifying x = 4.455467416Subproblem 2
x + 6.75 = -11.205467416 Simplifying x + 6.75 = -11.205467416 Reorder the terms: 6.75 + x = -11.205467416 Solving 6.75 + x = -11.205467416 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-6.75' to each side of the equation. 6.75 + -6.75 + x = -11.205467416 + -6.75 Combine like terms: 6.75 + -6.75 = 0.00 0.00 + x = -11.205467416 + -6.75 x = -11.205467416 + -6.75 Combine like terms: -11.205467416 + -6.75 = -17.955467416 x = -17.955467416 Simplifying x = -17.955467416Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.455467416, -17.955467416}Solution
x = {4.455467416, -17.955467416}
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