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Simplifying (x + 8)(x + 120) = 16500 Reorder the terms: (8 + x)(x + 120) = 16500 Reorder the terms: (8 + x)(120 + x) = 16500 Multiply (8 + x) * (120 + x) (8(120 + x) + x(120 + x)) = 16500 ((120 * 8 + x * 8) + x(120 + x)) = 16500 ((960 + 8x) + x(120 + x)) = 16500 (960 + 8x + (120 * x + x * x)) = 16500 (960 + 8x + (120x + x2)) = 16500 Combine like terms: 8x + 120x = 128x (960 + 128x + x2) = 16500 Solving 960 + 128x + x2 = 16500 Solving for variable 'x'. Reorder the terms: 960 + -16500 + 128x + x2 = 16500 + -16500 Combine like terms: 960 + -16500 = -15540 -15540 + 128x + x2 = 16500 + -16500 Combine like terms: 16500 + -16500 = 0 -15540 + 128x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '15540' to each side of the equation. -15540 + 128x + 15540 + x2 = 0 + 15540 Reorder the terms: -15540 + 15540 + 128x + x2 = 0 + 15540 Combine like terms: -15540 + 15540 = 0 0 + 128x + x2 = 0 + 15540 128x + x2 = 0 + 15540 Combine like terms: 0 + 15540 = 15540 128x + x2 = 15540 The x term is 128x. Take half its coefficient (64). Square it (4096) and add it to both sides. Add '4096' to each side of the equation. 128x + 4096 + x2 = 15540 + 4096 Reorder the terms: 4096 + 128x + x2 = 15540 + 4096 Combine like terms: 15540 + 4096 = 19636 4096 + 128x + x2 = 19636 Factor a perfect square on the left side: (x + 64)(x + 64) = 19636 Calculate the square root of the right side: 140.128512445 Break this problem into two subproblems by setting (x + 64) equal to 140.128512445 and -140.128512445.Subproblem 1
x + 64 = 140.128512445 Simplifying x + 64 = 140.128512445 Reorder the terms: 64 + x = 140.128512445 Solving 64 + x = 140.128512445 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-64' to each side of the equation. 64 + -64 + x = 140.128512445 + -64 Combine like terms: 64 + -64 = 0 0 + x = 140.128512445 + -64 x = 140.128512445 + -64 Combine like terms: 140.128512445 + -64 = 76.128512445 x = 76.128512445 Simplifying x = 76.128512445Subproblem 2
x + 64 = -140.128512445 Simplifying x + 64 = -140.128512445 Reorder the terms: 64 + x = -140.128512445 Solving 64 + x = -140.128512445 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-64' to each side of the equation. 64 + -64 + x = -140.128512445 + -64 Combine like terms: 64 + -64 = 0 0 + x = -140.128512445 + -64 x = -140.128512445 + -64 Combine like terms: -140.128512445 + -64 = -204.128512445 x = -204.128512445 Simplifying x = -204.128512445Solution
The solution to the problem is based on the solutions from the subproblems. x = {76.128512445, -204.128512445}
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