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Simplifying x(2x + 3) = 216 Reorder the terms: x(3 + 2x) = 216 (3 * x + 2x * x) = 216 (3x + 2x2) = 216 Solving 3x + 2x2 = 216 Solving for variable 'x'. Reorder the terms: -216 + 3x + 2x2 = 216 + -216 Combine like terms: 216 + -216 = 0 -216 + 3x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -108 + 1.5x + x2 = 0 Move the constant term to the right: Add '108' to each side of the equation. -108 + 1.5x + 108 + x2 = 0 + 108 Reorder the terms: -108 + 108 + 1.5x + x2 = 0 + 108 Combine like terms: -108 + 108 = 0 0 + 1.5x + x2 = 0 + 108 1.5x + x2 = 0 + 108 Combine like terms: 0 + 108 = 108 1.5x + x2 = 108 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 = 108 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 108 + 0.5625 Combine like terms: 108 + 0.5625 = 108.5625 0.5625 + 1.5x + x2 = 108.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 108.5625 Calculate the square root of the right side: 10.419332992 Break this problem into two subproblems by setting (x + 0.75) equal to 10.419332992 and -10.419332992.Subproblem 1
x + 0.75 = 10.419332992 Simplifying x + 0.75 = 10.419332992 Reorder the terms: 0.75 + x = 10.419332992 Solving 0.75 + x = 10.419332992 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 = 10.419332992 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 10.419332992 + -0.75 x = 10.419332992 + -0.75 Combine like terms: 10.419332992 + -0.75 = 9.669332992 x = 9.669332992 Simplifying x = 9.669332992Subproblem 2
x + 0.75 = -10.419332992 Simplifying x + 0.75 = -10.419332992 Reorder the terms: 0.75 + x = -10.419332992 Solving 0.75 + x = -10.419332992 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 = -10.419332992 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -10.419332992 + -0.75 x = -10.419332992 + -0.75 Combine like terms: -10.419332992 + -0.75 = -11.169332992 x = -11.169332992 Simplifying x = -11.169332992Solution
The solution to the problem is based on the solutions from the subproblems. x = {9.669332992, -11.169332992}
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