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Simplifying 2x2 + 3x + -108 = 0 Reorder the terms: -108 + 3x + 2x2 = 0 Solving -108 + 3x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -54 + 1.5x + x2 = 0 Move the constant term to the right: Add '54' to each side of the equation. -54 + 1.5x + 54 + x2 = 0 + 54 Reorder the terms: -54 + 54 + 1.5x + x2 = 0 + 54 Combine like terms: -54 + 54 = 0 0 + 1.5x + x2 = 0 + 54 1.5x + x2 = 0 + 54 Combine like terms: 0 + 54 = 54 1.5x + x2 = 54 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 = 54 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 54 + 0.5625 Combine like terms: 54 + 0.5625 = 54.5625 0.5625 + 1.5x + x2 = 54.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 54.5625 Calculate the square root of the right side: 7.386643351 Break this problem into two subproblems by setting (x + 0.75) equal to 7.386643351 and -7.386643351.Subproblem 1
x + 0.75 = 7.386643351 Simplifying x + 0.75 = 7.386643351 Reorder the terms: 0.75 + x = 7.386643351 Solving 0.75 + x = 7.386643351 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 = 7.386643351 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 7.386643351 + -0.75 x = 7.386643351 + -0.75 Combine like terms: 7.386643351 + -0.75 = 6.636643351 x = 6.636643351 Simplifying x = 6.636643351Subproblem 2
x + 0.75 = -7.386643351 Simplifying x + 0.75 = -7.386643351 Reorder the terms: 0.75 + x = -7.386643351 Solving 0.75 + x = -7.386643351 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 = -7.386643351 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -7.386643351 + -0.75 x = -7.386643351 + -0.75 Combine like terms: -7.386643351 + -0.75 = -8.136643351 x = -8.136643351 Simplifying x = -8.136643351Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.636643351, -8.136643351}
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