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Simplifying 128x2 + 16x + -14 = 0 Reorder the terms: -14 + 16x + 128x2 = 0 Solving -14 + 16x + 128x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + 8x + 64x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + 8x + 64x2)' equal to zero and attempt to solve: Simplifying -7 + 8x + 64x2 = 0 Solving -7 + 8x + 64x2 = 0 Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. -0.109375 + 0.125x + x2 = 0 Move the constant term to the right: Add '0.109375' to each side of the equation. -0.109375 + 0.125x + 0.109375 + x2 = 0 + 0.109375 Reorder the terms: -0.109375 + 0.109375 + 0.125x + x2 = 0 + 0.109375 Combine like terms: -0.109375 + 0.109375 = 0.000000 0.000000 + 0.125x + x2 = 0 + 0.109375 0.125x + x2 = 0 + 0.109375 Combine like terms: 0 + 0.109375 = 0.109375 0.125x + x2 = 0.109375 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 = 0.109375 + 0.00390625 Reorder the terms: 0.00390625 + 0.125x + x2 = 0.109375 + 0.00390625 Combine like terms: 0.109375 + 0.00390625 = 0.11328125 0.00390625 + 0.125x + x2 = 0.11328125 Factor a perfect square on the left side: (x + 0.0625)(x + 0.0625) = 0.11328125 Calculate the square root of the right side: 0.3365728 Break this problem into two subproblems by setting (x + 0.0625) equal to 0.3365728 and -0.3365728.Subproblem 1
x + 0.0625 = 0.3365728 Simplifying x + 0.0625 = 0.3365728 Reorder the terms: 0.0625 + x = 0.3365728 Solving 0.0625 + x = 0.3365728 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 = 0.3365728 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + x = 0.3365728 + -0.0625 x = 0.3365728 + -0.0625 Combine like terms: 0.3365728 + -0.0625 = 0.2740728 x = 0.2740728 Simplifying x = 0.2740728Subproblem 2
x + 0.0625 = -0.3365728 Simplifying x + 0.0625 = -0.3365728 Reorder the terms: 0.0625 + x = -0.3365728 Solving 0.0625 + x = -0.3365728 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 = -0.3365728 + -0.0625 Combine like terms: 0.0625 + -0.0625 = 0.0000 0.0000 + x = -0.3365728 + -0.0625 x = -0.3365728 + -0.0625 Combine like terms: -0.3365728 + -0.0625 = -0.3990728 x = -0.3990728 Simplifying x = -0.3990728Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.2740728, -0.3990728}Solution
x = {0.2740728, -0.3990728}
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