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Simplifying 16x2 + 14x + -214 = 0 Reorder the terms: -214 + 14x + 16x2 = 0 Solving -214 + 14x + 16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-107 + 7x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-107 + 7x + 8x2)' equal to zero and attempt to solve: Simplifying -107 + 7x + 8x2 = 0 Solving -107 + 7x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -13.375 + 0.875x + x2 = 0 Move the constant term to the right: Add '13.375' to each side of the equation. -13.375 + 0.875x + 13.375 + x2 = 0 + 13.375 Reorder the terms: -13.375 + 13.375 + 0.875x + x2 = 0 + 13.375 Combine like terms: -13.375 + 13.375 = 0.000 0.000 + 0.875x + x2 = 0 + 13.375 0.875x + x2 = 0 + 13.375 Combine like terms: 0 + 13.375 = 13.375 0.875x + x2 = 13.375 The x term is 0.875x. Take half its coefficient (0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. 0.875x + 0.19140625 + x2 = 13.375 + 0.19140625 Reorder the terms: 0.19140625 + 0.875x + x2 = 13.375 + 0.19140625 Combine like terms: 13.375 + 0.19140625 = 13.56640625 0.19140625 + 0.875x + x2 = 13.56640625 Factor a perfect square on the left side: (x + 0.4375)(x + 0.4375) = 13.56640625 Calculate the square root of the right side: 3.683260275 Break this problem into two subproblems by setting (x + 0.4375) equal to 3.683260275 and -3.683260275.Subproblem 1
x + 0.4375 = 3.683260275 Simplifying x + 0.4375 = 3.683260275 Reorder the terms: 0.4375 + x = 3.683260275 Solving 0.4375 + x = 3.683260275 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + x = 3.683260275 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + x = 3.683260275 + -0.4375 x = 3.683260275 + -0.4375 Combine like terms: 3.683260275 + -0.4375 = 3.245760275 x = 3.245760275 Simplifying x = 3.245760275Subproblem 2
x + 0.4375 = -3.683260275 Simplifying x + 0.4375 = -3.683260275 Reorder the terms: 0.4375 + x = -3.683260275 Solving 0.4375 + x = -3.683260275 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + x = -3.683260275 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + x = -3.683260275 + -0.4375 x = -3.683260275 + -0.4375 Combine like terms: -3.683260275 + -0.4375 = -4.120760275 x = -4.120760275 Simplifying x = -4.120760275Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.245760275, -4.120760275}Solution
x = {3.245760275, -4.120760275}
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