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Simplifying -16x2 + -114x + 14 = 0 Reorder the terms: 14 + -114x + -16x2 = 0 Solving 14 + -114x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(7 + -57x + -8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(7 + -57x + -8x2)' equal to zero and attempt to solve: Simplifying 7 + -57x + -8x2 = 0 Solving 7 + -57x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.875 + 7.125x + x2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + 7.125x + 0.875 + x2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + 7.125x + x2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + 7.125x + x2 = 0 + 0.875 7.125x + x2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 7.125x + x2 = 0.875 The x term is 7.125x. Take half its coefficient (3.5625). Square it (12.69140625) and add it to both sides. Add '12.69140625' to each side of the equation. 7.125x + 12.69140625 + x2 = 0.875 + 12.69140625 Reorder the terms: 12.69140625 + 7.125x + x2 = 0.875 + 12.69140625 Combine like terms: 0.875 + 12.69140625 = 13.56640625 12.69140625 + 7.125x + x2 = 13.56640625 Factor a perfect square on the left side: (x + 3.5625)(x + 3.5625) = 13.56640625 Calculate the square root of the right side: 3.683260275 Break this problem into two subproblems by setting (x + 3.5625) equal to 3.683260275 and -3.683260275.Subproblem 1
x + 3.5625 = 3.683260275 Simplifying x + 3.5625 = 3.683260275 Reorder the terms: 3.5625 + x = 3.683260275 Solving 3.5625 + x = 3.683260275 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5625' to each side of the equation. 3.5625 + -3.5625 + x = 3.683260275 + -3.5625 Combine like terms: 3.5625 + -3.5625 = 0.0000 0.0000 + x = 3.683260275 + -3.5625 x = 3.683260275 + -3.5625 Combine like terms: 3.683260275 + -3.5625 = 0.120760275 x = 0.120760275 Simplifying x = 0.120760275Subproblem 2
x + 3.5625 = -3.683260275 Simplifying x + 3.5625 = -3.683260275 Reorder the terms: 3.5625 + x = -3.683260275 Solving 3.5625 + x = -3.683260275 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-3.5625' to each side of the equation. 3.5625 + -3.5625 + x = -3.683260275 + -3.5625 Combine like terms: 3.5625 + -3.5625 = 0.0000 0.0000 + x = -3.683260275 + -3.5625 x = -3.683260275 + -3.5625 Combine like terms: -3.683260275 + -3.5625 = -7.245760275 x = -7.245760275 Simplifying x = -7.245760275Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.120760275, -7.245760275}Solution
x = {0.120760275, -7.245760275}
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