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Simplifying -16x2 + 32x + -8 = 0 Reorder the terms: -8 + 32x + -16x2 = 0 Solving -8 + 32x + -16x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '8'. 8(-1 + 4x + -2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(-1 + 4x + -2x2)' equal to zero and attempt to solve: Simplifying -1 + 4x + -2x2 = 0 Solving -1 + 4x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 0.5 + -2x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2x + x2 = 0 + -0.5 -2x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2x + x2 = -0.5 The x term is -2x. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2x + 1 + x2 = -0.5 + 1 Reorder the terms: 1 + -2x + x2 = -0.5 + 1 Combine like terms: -0.5 + 1 = 0.5 1 + -2x + x2 = 0.5 Factor a perfect square on the left side: (x + -1)(x + -1) = 0.5 Calculate the square root of the right side: 0.707106781 Break this problem into two subproblems by setting (x + -1) equal to 0.707106781 and -0.707106781.Subproblem 1
x + -1 = 0.707106781 Simplifying x + -1 = 0.707106781 Reorder the terms: -1 + x = 0.707106781 Solving -1 + x = 0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = 0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x = 0.707106781 + 1 x = 0.707106781 + 1 Combine like terms: 0.707106781 + 1 = 1.707106781 x = 1.707106781 Simplifying x = 1.707106781Subproblem 2
x + -1 = -0.707106781 Simplifying x + -1 = -0.707106781 Reorder the terms: -1 + x = -0.707106781 Solving -1 + x = -0.707106781 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + x = -0.707106781 + 1 Combine like terms: -1 + 1 = 0 0 + x = -0.707106781 + 1 x = -0.707106781 + 1 Combine like terms: -0.707106781 + 1 = 0.292893219 x = 0.292893219 Simplifying x = 0.292893219Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.707106781, 0.292893219}Solution
x = {1.707106781, 0.292893219}
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