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Simplifying (X + 5)(X + 2) = 117 Reorder the terms: (5 + X)(X + 2) = 117 Reorder the terms: (5 + X)(2 + X) = 117 Multiply (5 + X) * (2 + X) (5(2 + X) + X(2 + X)) = 117 ((2 * 5 + X * 5) + X(2 + X)) = 117 ((10 + 5X) + X(2 + X)) = 117 (10 + 5X + (2 * X + X * X)) = 117 (10 + 5X + (2X + X2)) = 117 Combine like terms: 5X + 2X = 7X (10 + 7X + X2) = 117 Solving 10 + 7X + X2 = 117 Solving for variable 'X'. Reorder the terms: 10 + -117 + 7X + X2 = 117 + -117 Combine like terms: 10 + -117 = -107 -107 + 7X + X2 = 117 + -117 Combine like terms: 117 + -117 = 0 -107 + 7X + X2 = 0 Begin completing the square. Move the constant term to the right: Add '107' to each side of the equation. -107 + 7X + 107 + X2 = 0 + 107 Reorder the terms: -107 + 107 + 7X + X2 = 0 + 107 Combine like terms: -107 + 107 = 0 0 + 7X + X2 = 0 + 107 7X + X2 = 0 + 107 Combine like terms: 0 + 107 = 107 7X + X2 = 107 The X term is 7X. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7X + 12.25 + X2 = 107 + 12.25 Reorder the terms: 12.25 + 7X + X2 = 107 + 12.25 Combine like terms: 107 + 12.25 = 119.25 12.25 + 7X + X2 = 119.25 Factor a perfect square on the left side: (X + 3.5)(X + 3.5) = 119.25 Calculate the square root of the right side: 10.920164834 Break this problem into two subproblems by setting (X + 3.5) equal to 10.920164834 and -10.920164834.Subproblem 1
X + 3.5 = 10.920164834 Simplifying X + 3.5 = 10.920164834 Reorder the terms: 3.5 + X = 10.920164834 Solving 3.5 + X = 10.920164834 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + X = 10.920164834 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + X = 10.920164834 + -3.5 X = 10.920164834 + -3.5 Combine like terms: 10.920164834 + -3.5 = 7.420164834 X = 7.420164834 Simplifying X = 7.420164834Subproblem 2
X + 3.5 = -10.920164834 Simplifying X + 3.5 = -10.920164834 Reorder the terms: 3.5 + X = -10.920164834 Solving 3.5 + X = -10.920164834 Solving for variable 'X'. Move all terms containing X to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + X = -10.920164834 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + X = -10.920164834 + -3.5 X = -10.920164834 + -3.5 Combine like terms: -10.920164834 + -3.5 = -14.420164834 X = -14.420164834 Simplifying X = -14.420164834Solution
The solution to the problem is based on the solutions from the subproblems. X = {7.420164834, -14.420164834}
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