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Simplifying (2x + 3) * x = 180 Reorder the terms: (3 + 2x) * x = 180 Reorder the terms for easier multiplication: x(3 + 2x) = 180 (3 * x + 2x * x) = 180 (3x + 2x2) = 180 Solving 3x + 2x2 = 180 Solving for variable 'x'. Reorder the terms: -180 + 3x + 2x2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -90 + 1.5x + x2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + 1.5x + 90 + x2 = 0 + 90 Reorder the terms: -90 + 90 + 1.5x + x2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + 1.5x + x2 = 0 + 90 1.5x + x2 = 0 + 90 Combine like terms: 0 + 90 = 90 1.5x + x2 = 90 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 90 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 90 + 0.5625 Combine like terms: 90 + 0.5625 = 90.5625 0.5625 + 1.5x + x2 = 90.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 90.5625 Calculate the square root of the right side: 9.516433155 Break this problem into two subproblems by setting (x + 0.75) equal to 9.516433155 and -9.516433155.Subproblem 1
x + 0.75 = 9.516433155 Simplifying x + 0.75 = 9.516433155 Reorder the terms: 0.75 + x = 9.516433155 Solving 0.75 + x = 9.516433155 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 9.516433155 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 9.516433155 + -0.75 x = 9.516433155 + -0.75 Combine like terms: 9.516433155 + -0.75 = 8.766433155 x = 8.766433155 Simplifying x = 8.766433155Subproblem 2
x + 0.75 = -9.516433155 Simplifying x + 0.75 = -9.516433155 Reorder the terms: 0.75 + x = -9.516433155 Solving 0.75 + x = -9.516433155 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -9.516433155 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -9.516433155 + -0.75 x = -9.516433155 + -0.75 Combine like terms: -9.516433155 + -0.75 = -10.266433155 x = -10.266433155 Simplifying x = -10.266433155Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.766433155, -10.266433155}
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