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Simplifying (x + 10)(x + 5) + x = 180 Reorder the terms: (10 + x)(x + 5) + x = 180 Reorder the terms: (10 + x)(5 + x) + x = 180 Multiply (10 + x) * (5 + x) (10(5 + x) + x(5 + x)) + x = 180 ((5 * 10 + x * 10) + x(5 + x)) + x = 180 ((50 + 10x) + x(5 + x)) + x = 180 (50 + 10x + (5 * x + x * x)) + x = 180 (50 + 10x + (5x + x2)) + x = 180 Combine like terms: 10x + 5x = 15x (50 + 15x + x2) + x = 180 Reorder the terms: 50 + 15x + x + x2 = 180 Combine like terms: 15x + x = 16x 50 + 16x + x2 = 180 Solving 50 + 16x + x2 = 180 Solving for variable 'x'. Reorder the terms: 50 + -180 + 16x + x2 = 180 + -180 Combine like terms: 50 + -180 = -130 -130 + 16x + x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -130 + 16x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '130' to each side of the equation. -130 + 16x + 130 + x2 = 0 + 130 Reorder the terms: -130 + 130 + 16x + x2 = 0 + 130 Combine like terms: -130 + 130 = 0 0 + 16x + x2 = 0 + 130 16x + x2 = 0 + 130 Combine like terms: 0 + 130 = 130 16x + x2 = 130 The x term is 16x. Take half its coefficient (8). Square it (64) and add it to both sides. Add '64' to each side of the equation. 16x + 64 + x2 = 130 + 64 Reorder the terms: 64 + 16x + x2 = 130 + 64 Combine like terms: 130 + 64 = 194 64 + 16x + x2 = 194 Factor a perfect square on the left side: (x + 8)(x + 8) = 194 Calculate the square root of the right side: 13.928388277 Break this problem into two subproblems by setting (x + 8) equal to 13.928388277 and -13.928388277.Subproblem 1
x + 8 = 13.928388277 Simplifying x + 8 = 13.928388277 Reorder the terms: 8 + x = 13.928388277 Solving 8 + x = 13.928388277 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + x = 13.928388277 + -8 Combine like terms: 8 + -8 = 0 0 + x = 13.928388277 + -8 x = 13.928388277 + -8 Combine like terms: 13.928388277 + -8 = 5.928388277 x = 5.928388277 Simplifying x = 5.928388277Subproblem 2
x + 8 = -13.928388277 Simplifying x + 8 = -13.928388277 Reorder the terms: 8 + x = -13.928388277 Solving 8 + x = -13.928388277 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-8' to each side of the equation. 8 + -8 + x = -13.928388277 + -8 Combine like terms: 8 + -8 = 0 0 + x = -13.928388277 + -8 x = -13.928388277 + -8 Combine like terms: -13.928388277 + -8 = -21.928388277 x = -21.928388277 Simplifying x = -21.928388277Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.928388277, -21.928388277}
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