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Simplifying (2x + 5) * x = 82 Reorder the terms: (5 + 2x) * x = 82 Reorder the terms for easier multiplication: x(5 + 2x) = 82 (5 * x + 2x * x) = 82 (5x + 2x2) = 82 Solving 5x + 2x2 = 82 Solving for variable 'x'. Reorder the terms: -82 + 5x + 2x2 = 82 + -82 Combine like terms: 82 + -82 = 0 -82 + 5x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -41 + 2.5x + x2 = 0 Move the constant term to the right: Add '41' to each side of the equation. -41 + 2.5x + 41 + x2 = 0 + 41 Reorder the terms: -41 + 41 + 2.5x + x2 = 0 + 41 Combine like terms: -41 + 41 = 0 0 + 2.5x + x2 = 0 + 41 2.5x + x2 = 0 + 41 Combine like terms: 0 + 41 = 41 2.5x + x2 = 41 The x term is 2.5x. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5x + 1.5625 + x2 = 41 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = 41 + 1.5625 Combine like terms: 41 + 1.5625 = 42.5625 1.5625 + 2.5x + x2 = 42.5625 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 42.5625 Calculate the square root of the right side: 6.523994175 Break this problem into two subproblems by setting (x + 1.25) equal to 6.523994175 and -6.523994175.Subproblem 1
x + 1.25 = 6.523994175 Simplifying x + 1.25 = 6.523994175 Reorder the terms: 1.25 + x = 6.523994175 Solving 1.25 + x = 6.523994175 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = 6.523994175 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 6.523994175 + -1.25 x = 6.523994175 + -1.25 Combine like terms: 6.523994175 + -1.25 = 5.273994175 x = 5.273994175 Simplifying x = 5.273994175Subproblem 2
x + 1.25 = -6.523994175 Simplifying x + 1.25 = -6.523994175 Reorder the terms: 1.25 + x = -6.523994175 Solving 1.25 + x = -6.523994175 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + x = -6.523994175 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -6.523994175 + -1.25 x = -6.523994175 + -1.25 Combine like terms: -6.523994175 + -1.25 = -7.773994175 x = -7.773994175 Simplifying x = -7.773994175Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.273994175, -7.773994175}
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