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Simplifying 8x2 + 27x + -58.5 = 0 Reorder the terms: -58.5 + 27x + 8x2 = 0 Solving -58.5 + 27x + 8x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -7.3125 + 3.375x + x2 = 0 Move the constant term to the right: Add '7.3125' to each side of the equation. -7.3125 + 3.375x + 7.3125 + x2 = 0 + 7.3125 Reorder the terms: -7.3125 + 7.3125 + 3.375x + x2 = 0 + 7.3125 Combine like terms: -7.3125 + 7.3125 = 0.0000 0.0000 + 3.375x + x2 = 0 + 7.3125 3.375x + x2 = 0 + 7.3125 Combine like terms: 0 + 7.3125 = 7.3125 3.375x + x2 = 7.3125 The x term is 3.375x. Take half its coefficient (1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. 3.375x + 2.84765625 + x2 = 7.3125 + 2.84765625 Reorder the terms: 2.84765625 + 3.375x + x2 = 7.3125 + 2.84765625 Combine like terms: 7.3125 + 2.84765625 = 10.16015625 2.84765625 + 3.375x + x2 = 10.16015625 Factor a perfect square on the left side: (x + 1.6875)(x + 1.6875) = 10.16015625 Calculate the square root of the right side: 3.1875 Break this problem into two subproblems by setting (x + 1.6875) equal to 3.1875 and -3.1875.Subproblem 1
x + 1.6875 = 3.1875 Simplifying x + 1.6875 = 3.1875 Reorder the terms: 1.6875 + x = 3.1875 Solving 1.6875 + x = 3.1875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + x = 3.1875 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + x = 3.1875 + -1.6875 x = 3.1875 + -1.6875 Combine like terms: 3.1875 + -1.6875 = 1.5 x = 1.5 Simplifying x = 1.5Subproblem 2
x + 1.6875 = -3.1875 Simplifying x + 1.6875 = -3.1875 Reorder the terms: 1.6875 + x = -3.1875 Solving 1.6875 + x = -3.1875 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.6875' to each side of the equation. 1.6875 + -1.6875 + x = -3.1875 + -1.6875 Combine like terms: 1.6875 + -1.6875 = 0.0000 0.0000 + x = -3.1875 + -1.6875 x = -3.1875 + -1.6875 Combine like terms: -3.1875 + -1.6875 = -4.875 x = -4.875 Simplifying x = -4.875Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5, -4.875}
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