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Simplifying (100 + x)(50 + x) = 84000 Multiply (100 + x) * (50 + x) (100(50 + x) + x(50 + x)) = 84000 ((50 * 100 + x * 100) + x(50 + x)) = 84000 ((5000 + 100x) + x(50 + x)) = 84000 (5000 + 100x + (50 * x + x * x)) = 84000 (5000 + 100x + (50x + x2)) = 84000 Combine like terms: 100x + 50x = 150x (5000 + 150x + x2) = 84000 Solving 5000 + 150x + x2 = 84000 Solving for variable 'x'. Reorder the terms: 5000 + -84000 + 150x + x2 = 84000 + -84000 Combine like terms: 5000 + -84000 = -79000 -79000 + 150x + x2 = 84000 + -84000 Combine like terms: 84000 + -84000 = 0 -79000 + 150x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '79000' to each side of the equation. -79000 + 150x + 79000 + x2 = 0 + 79000 Reorder the terms: -79000 + 79000 + 150x + x2 = 0 + 79000 Combine like terms: -79000 + 79000 = 0 0 + 150x + x2 = 0 + 79000 150x + x2 = 0 + 79000 Combine like terms: 0 + 79000 = 79000 150x + x2 = 79000 The x term is 150x. Take half its coefficient (75). Square it (5625) and add it to both sides. Add '5625' to each side of the equation. 150x + 5625 + x2 = 79000 + 5625 Reorder the terms: 5625 + 150x + x2 = 79000 + 5625 Combine like terms: 79000 + 5625 = 84625 5625 + 150x + x2 = 84625 Factor a perfect square on the left side: (x + 75)(x + 75) = 84625 Calculate the square root of the right side: 290.903764156 Break this problem into two subproblems by setting (x + 75) equal to 290.903764156 and -290.903764156.Subproblem 1
x + 75 = 290.903764156 Simplifying x + 75 = 290.903764156 Reorder the terms: 75 + x = 290.903764156 Solving 75 + x = 290.903764156 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-75' to each side of the equation. 75 + -75 + x = 290.903764156 + -75 Combine like terms: 75 + -75 = 0 0 + x = 290.903764156 + -75 x = 290.903764156 + -75 Combine like terms: 290.903764156 + -75 = 215.903764156 x = 215.903764156 Simplifying x = 215.903764156Subproblem 2
x + 75 = -290.903764156 Simplifying x + 75 = -290.903764156 Reorder the terms: 75 + x = -290.903764156 Solving 75 + x = -290.903764156 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-75' to each side of the equation. 75 + -75 + x = -290.903764156 + -75 Combine like terms: 75 + -75 = 0 0 + x = -290.903764156 + -75 x = -290.903764156 + -75 Combine like terms: -290.903764156 + -75 = -365.903764156 x = -365.903764156 Simplifying x = -365.903764156Solution
The solution to the problem is based on the solutions from the subproblems. x = {215.903764156, -365.903764156}
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