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Simplifying 64x2 + 32x + -108 = 0 Reorder the terms: -108 + 32x + 64x2 = 0 Solving -108 + 32x + 64x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '4'. 4(-27 + 8x + 16x2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-27 + 8x + 16x2)' equal to zero and attempt to solve: Simplifying -27 + 8x + 16x2 = 0 Solving -27 + 8x + 16x2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -1.6875 + 0.5x + x2 = 0 Move the constant term to the right: Add '1.6875' to each side of the equation. -1.6875 + 0.5x + 1.6875 + x2 = 0 + 1.6875 Reorder the terms: -1.6875 + 1.6875 + 0.5x + x2 = 0 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + 0.5x + x2 = 0 + 1.6875 0.5x + x2 = 0 + 1.6875 Combine like terms: 0 + 1.6875 = 1.6875 0.5x + x2 = 1.6875 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 1.6875 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 1.6875 + 0.0625 Combine like terms: 1.6875 + 0.0625 = 1.75 0.0625 + 0.5x + x2 = 1.75 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (x + 0.25) equal to 1.322875656 and -1.322875656.Subproblem 1
x + 0.25 = 1.322875656 Simplifying x + 0.25 = 1.322875656 Reorder the terms: 0.25 + x = 1.322875656 Solving 0.25 + x = 1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 1.322875656 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 1.322875656 + -0.25 x = 1.322875656 + -0.25 Combine like terms: 1.322875656 + -0.25 = 1.072875656 x = 1.072875656 Simplifying x = 1.072875656Subproblem 2
x + 0.25 = -1.322875656 Simplifying x + 0.25 = -1.322875656 Reorder the terms: 0.25 + x = -1.322875656 Solving 0.25 + x = -1.322875656 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -1.322875656 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -1.322875656 + -0.25 x = -1.322875656 + -0.25 Combine like terms: -1.322875656 + -0.25 = -1.572875656 x = -1.572875656 Simplifying x = -1.572875656Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.072875656, -1.572875656}Solution
x = {1.072875656, -1.572875656}
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