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Simplifying 24x2 + 120x + -185 = 0 Reorder the terms: -185 + 120x + 24x2 = 0 Solving -185 + 120x + 24x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -7.708333333 + 5x + x2 = 0 Move the constant term to the right: Add '7.708333333' to each side of the equation. -7.708333333 + 5x + 7.708333333 + x2 = 0 + 7.708333333 Reorder the terms: -7.708333333 + 7.708333333 + 5x + x2 = 0 + 7.708333333 Combine like terms: -7.708333333 + 7.708333333 = 0.000000000 0.000000000 + 5x + x2 = 0 + 7.708333333 5x + x2 = 0 + 7.708333333 Combine like terms: 0 + 7.708333333 = 7.708333333 5x + x2 = 7.708333333 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 7.708333333 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 7.708333333 + 6.25 Combine like terms: 7.708333333 + 6.25 = 13.958333333 6.25 + 5x + x2 = 13.958333333 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 13.958333333 Calculate the square root of the right side: 3.736085295 Break this problem into two subproblems by setting (x + 2.5) equal to 3.736085295 and -3.736085295.Subproblem 1
x + 2.5 = 3.736085295 Simplifying x + 2.5 = 3.736085295 Reorder the terms: 2.5 + x = 3.736085295 Solving 2.5 + x = 3.736085295 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 3.736085295 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 3.736085295 + -2.5 x = 3.736085295 + -2.5 Combine like terms: 3.736085295 + -2.5 = 1.236085295 x = 1.236085295 Simplifying x = 1.236085295Subproblem 2
x + 2.5 = -3.736085295 Simplifying x + 2.5 = -3.736085295 Reorder the terms: 2.5 + x = -3.736085295 Solving 2.5 + x = -3.736085295 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -3.736085295 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -3.736085295 + -2.5 x = -3.736085295 + -2.5 Combine like terms: -3.736085295 + -2.5 = -6.236085295 x = -6.236085295 Simplifying x = -6.236085295Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.236085295, -6.236085295}
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