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Simplifying (x + 20)(x + 30) = 1000 Reorder the terms: (20 + x)(x + 30) = 1000 Reorder the terms: (20 + x)(30 + x) = 1000 Multiply (20 + x) * (30 + x) (20(30 + x) + x(30 + x)) = 1000 ((30 * 20 + x * 20) + x(30 + x)) = 1000 ((600 + 20x) + x(30 + x)) = 1000 (600 + 20x + (30 * x + x * x)) = 1000 (600 + 20x + (30x + x2)) = 1000 Combine like terms: 20x + 30x = 50x (600 + 50x + x2) = 1000 Solving 600 + 50x + x2 = 1000 Solving for variable 'x'. Reorder the terms: 600 + -1000 + 50x + x2 = 1000 + -1000 Combine like terms: 600 + -1000 = -400 -400 + 50x + x2 = 1000 + -1000 Combine like terms: 1000 + -1000 = 0 -400 + 50x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '400' to each side of the equation. -400 + 50x + 400 + x2 = 0 + 400 Reorder the terms: -400 + 400 + 50x + x2 = 0 + 400 Combine like terms: -400 + 400 = 0 0 + 50x + x2 = 0 + 400 50x + x2 = 0 + 400 Combine like terms: 0 + 400 = 400 50x + x2 = 400 The x term is 50x. Take half its coefficient (25). Square it (625) and add it to both sides. Add '625' to each side of the equation. 50x + 625 + x2 = 400 + 625 Reorder the terms: 625 + 50x + x2 = 400 + 625 Combine like terms: 400 + 625 = 1025 625 + 50x + x2 = 1025 Factor a perfect square on the left side: (x + 25)(x + 25) = 1025 Calculate the square root of the right side: 32.015621187 Break this problem into two subproblems by setting (x + 25) equal to 32.015621187 and -32.015621187.Subproblem 1
x + 25 = 32.015621187 Simplifying x + 25 = 32.015621187 Reorder the terms: 25 + x = 32.015621187 Solving 25 + x = 32.015621187 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-25' to each side of the equation. 25 + -25 + x = 32.015621187 + -25 Combine like terms: 25 + -25 = 0 0 + x = 32.015621187 + -25 x = 32.015621187 + -25 Combine like terms: 32.015621187 + -25 = 7.015621187 x = 7.015621187 Simplifying x = 7.015621187Subproblem 2
x + 25 = -32.015621187 Simplifying x + 25 = -32.015621187 Reorder the terms: 25 + x = -32.015621187 Solving 25 + x = -32.015621187 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-25' to each side of the equation. 25 + -25 + x = -32.015621187 + -25 Combine like terms: 25 + -25 = 0 0 + x = -32.015621187 + -25 x = -32.015621187 + -25 Combine like terms: -32.015621187 + -25 = -57.015621187 x = -57.015621187 Simplifying x = -57.015621187Solution
The solution to the problem is based on the solutions from the subproblems. x = {7.015621187, -57.015621187}
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