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Simplifying (x + 2)(x + 3) = 84 Reorder the terms: (2 + x)(x + 3) = 84 Reorder the terms: (2 + x)(3 + x) = 84 Multiply (2 + x) * (3 + x) (2(3 + x) + x(3 + x)) = 84 ((3 * 2 + x * 2) + x(3 + x)) = 84 ((6 + 2x) + x(3 + x)) = 84 (6 + 2x + (3 * x + x * x)) = 84 (6 + 2x + (3x + x2)) = 84 Combine like terms: 2x + 3x = 5x (6 + 5x + x2) = 84 Solving 6 + 5x + x2 = 84 Solving for variable 'x'. Reorder the terms: 6 + -84 + 5x + x2 = 84 + -84 Combine like terms: 6 + -84 = -78 -78 + 5x + x2 = 84 + -84 Combine like terms: 84 + -84 = 0 -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}
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