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Simplifying 2(x + 9)(x + -4) = 84 Reorder the terms: 2(9 + x)(x + -4) = 84 Reorder the terms: 2(9 + x)(-4 + x) = 84 Multiply (9 + x) * (-4 + x) 2(9(-4 + x) + x(-4 + x)) = 84 2((-4 * 9 + x * 9) + x(-4 + x)) = 84 2((-36 + 9x) + x(-4 + x)) = 84 2(-36 + 9x + (-4 * x + x * x)) = 84 2(-36 + 9x + (-4x + x2)) = 84 Combine like terms: 9x + -4x = 5x 2(-36 + 5x + x2) = 84 (-36 * 2 + 5x * 2 + x2 * 2) = 84 (-72 + 10x + 2x2) = 84 Solving -72 + 10x + 2x2 = 84 Solving for variable 'x'. Reorder the terms: -72 + -84 + 10x + 2x2 = 84 + -84 Combine like terms: -72 + -84 = -156 -156 + 10x + 2x2 = 84 + -84 Combine like terms: 84 + -84 = 0 -156 + 10x + 2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-78 + 5x + x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-78 + 5x + x2)' equal to zero and attempt to solve: Simplifying -78 + 5x + x2 = 0 Solving -78 + 5x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '78' to each side of the equation. -78 + 5x + 78 + x2 = 0 + 78 Reorder the terms: -78 + 78 + 5x + x2 = 0 + 78 Combine like terms: -78 + 78 = 0 0 + 5x + x2 = 0 + 78 5x + x2 = 0 + 78 Combine like terms: 0 + 78 = 78 5x + x2 = 78 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 = 78 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 78 + 6.25 Combine like terms: 78 + 6.25 = 84.25 6.25 + 5x + x2 = 84.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 84.25 Calculate the square root of the right side: 9.178779875 Break this problem into two subproblems by setting (x + 2.5) equal to 9.178779875 and -9.178779875.Subproblem 1
x + 2.5 = 9.178779875 Simplifying x + 2.5 = 9.178779875 Reorder the terms: 2.5 + x = 9.178779875 Solving 2.5 + x = 9.178779875 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 = 9.178779875 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 9.178779875 + -2.5 x = 9.178779875 + -2.5 Combine like terms: 9.178779875 + -2.5 = 6.678779875 x = 6.678779875 Simplifying x = 6.678779875Subproblem 2
x + 2.5 = -9.178779875 Simplifying x + 2.5 = -9.178779875 Reorder the terms: 2.5 + x = -9.178779875 Solving 2.5 + x = -9.178779875 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 = -9.178779875 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -9.178779875 + -2.5 x = -9.178779875 + -2.5 Combine like terms: -9.178779875 + -2.5 = -11.678779875 x = -11.678779875 Simplifying x = -11.678779875Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.678779875, -11.678779875}Solution
x = {6.678779875, -11.678779875}
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