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Simplifying (2x + 4)(2x + -9) = 180 Reorder the terms: (4 + 2x)(2x + -9) = 180 Reorder the terms: (4 + 2x)(-9 + 2x) = 180 Multiply (4 + 2x) * (-9 + 2x) (4(-9 + 2x) + 2x * (-9 + 2x)) = 180 ((-9 * 4 + 2x * 4) + 2x * (-9 + 2x)) = 180 ((-36 + 8x) + 2x * (-9 + 2x)) = 180 (-36 + 8x + (-9 * 2x + 2x * 2x)) = 180 (-36 + 8x + (-18x + 4x2)) = 180 Combine like terms: 8x + -18x = -10x (-36 + -10x + 4x2) = 180 Solving -36 + -10x + 4x2 = 180 Solving for variable 'x'. Reorder the terms: -36 + -180 + -10x + 4x2 = 180 + -180 Combine like terms: -36 + -180 = -216 -216 + -10x + 4x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -216 + -10x + 4x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-108 + -5x + 2x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-108 + -5x + 2x2)' equal to zero and attempt to solve: Simplifying -108 + -5x + 2x2 = 0 Solving -108 + -5x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -54 + -2.5x + x2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + -2.5x + 54 + x2 = 0 + 54 Reorder the terms: -54 + 54 + -2.5x + x2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + -2.5x + x2 = 0 + 54 -2.5x + x2 = 0 + 54 Combine like terms: 0 + 54 = 54 -2.5x + x2 = 54 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = 54 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = 54 + 1.5625 Combine like terms: 54 + 1.5625 = 55.5625 1.5625 + -2.5x + x2 = 55.5625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 55.5625 Calculate the square root of the right side: 7.454025758 Break this problem into two subproblems by setting (x + -1.25) equal to 7.454025758 and -7.454025758.Subproblem 1
x + -1.25 = 7.454025758 Simplifying x + -1.25 = 7.454025758 Reorder the terms: -1.25 + x = 7.454025758 Solving -1.25 + x = 7.454025758 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 7.454025758 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 7.454025758 + 1.25 x = 7.454025758 + 1.25 Combine like terms: 7.454025758 + 1.25 = 8.704025758 x = 8.704025758 Simplifying x = 8.704025758Subproblem 2
x + -1.25 = -7.454025758 Simplifying x + -1.25 = -7.454025758 Reorder the terms: -1.25 + x = -7.454025758 Solving -1.25 + x = -7.454025758 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -7.454025758 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -7.454025758 + 1.25 x = -7.454025758 + 1.25 Combine like terms: -7.454025758 + 1.25 = -6.204025758 x = -6.204025758 Simplifying x = -6.204025758Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.704025758, -6.204025758}Solution
x = {8.704025758, -6.204025758}
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