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Simplifying x + x(x + 54) = 180 Reorder the terms: x + x(54 + x) = 180 x + (54 * x + x * x) = 180 x + (54x + x2) = 180 Combine like terms: x + 54x = 55x 55x + x2 = 180 Solving 55x + x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 55x + x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 55x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '180' to each side of the equation. -180 + 55x + 180 + x2 = 0 + 180 Reorder the terms: -180 + 180 + 55x + x2 = 0 + 180 Combine like terms: -180 + 180 = 0 0 + 55x + x2 = 0 + 180 55x + x2 = 0 + 180 Combine like terms: 0 + 180 = 180 55x + x2 = 180 The x term is 55x. Take half its coefficient (27.5). Square it (756.25) and add it to both sides. Add '756.25' to each side of the equation. 55x + 756.25 + x2 = 180 + 756.25 Reorder the terms: 756.25 + 55x + x2 = 180 + 756.25 Combine like terms: 180 + 756.25 = 936.25 756.25 + 55x + x2 = 936.25 Factor a perfect square on the left side: (x + 27.5)(x + 27.5) = 936.25 Calculate the square root of the right side: 30.598202562 Break this problem into two subproblems by setting (x + 27.5) equal to 30.598202562 and -30.598202562.Subproblem 1
x + 27.5 = 30.598202562 Simplifying x + 27.5 = 30.598202562 Reorder the terms: 27.5 + x = 30.598202562 Solving 27.5 + x = 30.598202562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-27.5' to each side of the equation. 27.5 + -27.5 + x = 30.598202562 + -27.5 Combine like terms: 27.5 + -27.5 = 0.0 0.0 + x = 30.598202562 + -27.5 x = 30.598202562 + -27.5 Combine like terms: 30.598202562 + -27.5 = 3.098202562 x = 3.098202562 Simplifying x = 3.098202562Subproblem 2
x + 27.5 = -30.598202562 Simplifying x + 27.5 = -30.598202562 Reorder the terms: 27.5 + x = -30.598202562 Solving 27.5 + x = -30.598202562 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-27.5' to each side of the equation. 27.5 + -27.5 + x = -30.598202562 + -27.5 Combine like terms: 27.5 + -27.5 = 0.0 0.0 + x = -30.598202562 + -27.5 x = -30.598202562 + -27.5 Combine like terms: -30.598202562 + -27.5 = -58.098202562 x = -58.098202562 Simplifying x = -58.098202562Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.098202562, -58.098202562}
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