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Simplifying (8x2 + -2x + -27) + (8x2 + 4x) = 0 Reorder the terms: (-27 + -2x + 8x2) + (8x2 + 4x) = 0 Remove parenthesis around (-27 + -2x + 8x2) -27 + -2x + 8x2 + (8x2 + 4x) = 0 Reorder the terms: -27 + -2x + 8x2 + (4x + 8x2) = 0 Remove parenthesis around (4x + 8x2) -27 + -2x + 8x2 + 4x + 8x2 = 0 Reorder the terms: -27 + -2x + 4x + 8x2 + 8x2 = 0 Combine like terms: -2x + 4x = 2x -27 + 2x + 8x2 + 8x2 = 0 Combine like terms: 8x2 + 8x2 = 16x2 -27 + 2x + 16x2 = 0 Solving -27 + 2x + 16x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.6875 + 0.125x + x2 = 0 Move the constant term to the right: Add '1.6875' to each side of the equation. -1.6875 + 0.125x + 1.6875 + x2 = 0 + 1.6875 Reorder the terms: -1.6875 + 1.6875 + 0.125x + x2 = 0 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + 0.125x + x2 = 0 + 1.6875 0.125x + x2 = 0 + 1.6875 Combine like terms: 0 + 1.6875 = 1.6875 0.125x + x2 = 1.6875 The x term is 0.125x. Take half its coefficient (0.0625). Square it (0.00390625) and add it to both sides. Add '0.00390625' to each side of the equation. 0.125x + 0.00390625 + x2 = 1.6875 + 0.00390625 Reorder the terms: 0.00390625 + 0.125x + x2 = 1.6875 + 0.00390625 Combine like terms: 1.6875 + 0.00390625 = 1.69140625 0.00390625 + 0.125x + x2 = 1.69140625 Factor a perfect square on the left side: (x + 0.0625)(x + 0.0625) = 1.69140625 Calculate the square root of the right side: 1.300540753 Break this problem into two subproblems by setting (x + 0.0625) equal to 1.300540753 and -1.300540753.Subproblem 1
x + 0.0625 = 1.300540753 Simplifying x + 0.0625 = 1.300540753 Reorder the terms: 0.0625 + x = 1.300540753 Solving 0.0625 + x = 1.300540753 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + x = 1.300540753 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + x = 1.300540753 + -0.0625 x = 1.300540753 + -0.0625 Combine like terms: 1.300540753 + -0.0625 = 1.238040753 x = 1.238040753 Simplifying x = 1.238040753Subproblem 2
x + 0.0625 = -1.300540753 Simplifying x + 0.0625 = -1.300540753 Reorder the terms: 0.0625 + x = -1.300540753 Solving 0.0625 + x = -1.300540753 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.0625' to each side of the equation. 0.0625 + -0.0625 + x = -1.300540753 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + x = -1.300540753 + -0.0625 x = -1.300540753 + -0.0625 Combine like terms: -1.300540753 + -0.0625 = -1.363040753 x = -1.363040753 Simplifying x = -1.363040753Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.238040753, -1.363040753}
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