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Simplifying (2x + 4)(2x + 6) = 500 Reorder the terms: (4 + 2x)(2x + 6) = 500 Reorder the terms: (4 + 2x)(6 + 2x) = 500 Multiply (4 + 2x) * (6 + 2x) (4(6 + 2x) + 2x * (6 + 2x)) = 500 ((6 * 4 + 2x * 4) + 2x * (6 + 2x)) = 500 ((24 + 8x) + 2x * (6 + 2x)) = 500 (24 + 8x + (6 * 2x + 2x * 2x)) = 500 (24 + 8x + (12x + 4x2)) = 500 Combine like terms: 8x + 12x = 20x (24 + 20x + 4x2) = 500 Solving 24 + 20x + 4x2 = 500 Solving for variable 'x'. Reorder the terms: 24 + -500 + 20x + 4x2 = 500 + -500 Combine like terms: 24 + -500 = -476 -476 + 20x + 4x2 = 500 + -500 Combine like terms: 500 + -500 = 0 -476 + 20x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-119 + 5x + x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-119 + 5x + x2)' equal to zero and attempt to solve: Simplifying -119 + 5x + x2 = 0 Solving -119 + 5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '119' to each side of the equation. -119 + 5x + 119 + x2 = 0 + 119 Reorder the terms: -119 + 119 + 5x + x2 = 0 + 119 Combine like terms: -119 + 119 = 0 0 + 5x + x2 = 0 + 119 5x + x2 = 0 + 119 Combine like terms: 0 + 119 = 119 5x + x2 = 119 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 119 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 119 + 6.25 Combine like terms: 119 + 6.25 = 125.25 6.25 + 5x + x2 = 125.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 125.25 Calculate the square root of the right side: 11.191514643 Break this problem into two subproblems by setting (x + 2.5) equal to 11.191514643 and -11.191514643.Subproblem 1
x + 2.5 = 11.191514643 Simplifying x + 2.5 = 11.191514643 Reorder the terms: 2.5 + x = 11.191514643 Solving 2.5 + x = 11.191514643 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 11.191514643 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 11.191514643 + -2.5 x = 11.191514643 + -2.5 Combine like terms: 11.191514643 + -2.5 = 8.691514643 x = 8.691514643 Simplifying x = 8.691514643Subproblem 2
x + 2.5 = -11.191514643 Simplifying x + 2.5 = -11.191514643 Reorder the terms: 2.5 + x = -11.191514643 Solving 2.5 + x = -11.191514643 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -11.191514643 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -11.191514643 + -2.5 x = -11.191514643 + -2.5 Combine like terms: -11.191514643 + -2.5 = -13.691514643 x = -13.691514643 Simplifying x = -13.691514643Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.691514643, -13.691514643}Solution
x = {8.691514643, -13.691514643}
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