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Simplifying (X + 3)(X + 2) = 84 Reorder the terms: (3 + X)(X + 2) = 84 Reorder the terms: (3 + X)(2 + X) = 84 Multiply (3 + X) * (2 + X) (3(2 + X) + X(2 + X)) = 84 ((2 * 3 + X * 3) + X(2 + X)) = 84 ((6 + 3X) + X(2 + X)) = 84 (6 + 3X + (2 * X + X * X)) = 84 (6 + 3X + (2X + X2)) = 84 Combine like terms: 3X + 2X = 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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