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Simplifying 0 = 32x + -16x2 + 2 Reorder the terms: 0 = 2 + 32x + -16x2 Solving 0 = 2 + 32x + -16x2 Solving for variable 'x'. Combine like terms: 0 + -2 = -2 -2 + -32x + 16x2 = 2 + 32x + -16x2 + -2 + -32x + 16x2 Reorder the terms: -2 + -32x + 16x2 = 2 + -2 + 32x + -32x + -16x2 + 16x2 Combine like terms: 2 + -2 = 0 -2 + -32x + 16x2 = 0 + 32x + -32x + -16x2 + 16x2 -2 + -32x + 16x2 = 32x + -32x + -16x2 + 16x2 Combine like terms: 32x + -32x = 0 -2 + -32x + 16x2 = 0 + -16x2 + 16x2 -2 + -32x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 -2 + -32x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-1 + -16x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-1 + -16x + 8x2)' equal to zero and attempt to solve: Simplifying -1 + -16x + 8x2 = 0 Solving -1 + -16x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.125 + -2x + x2 = 0 Move the constant term to the right: Add '0.125' to each side of the equation. -0.125 + -2x + 0.125 + x2 = 0 + 0.125 Reorder the terms: -0.125 + 0.125 + -2x + x2 = 0 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + -2x + x2 = 0 + 0.125 -2x + x2 = 0 + 0.125 Combine like terms: 0 + 0.125 = 0.125 -2x + x2 = 0.125 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 = 0.125 + 1 Reorder the terms: 1 + -2x + x2 = 0.125 + 1 Combine like terms: 0.125 + 1 = 1.125 1 + -2x + x2 = 1.125 Factor a perfect square on the left side: (x + -1)(x + -1) = 1.125 Calculate the square root of the right side: 1.060660172 Break this problem into two subproblems by setting (x + -1) equal to 1.060660172 and -1.060660172.Subproblem 1
x + -1 = 1.060660172 Simplifying x + -1 = 1.060660172 Reorder the terms: -1 + x = 1.060660172 Solving -1 + x = 1.060660172 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 = 1.060660172 + 1 Combine like terms: -1 + 1 = 0 0 + x = 1.060660172 + 1 x = 1.060660172 + 1 Combine like terms: 1.060660172 + 1 = 2.060660172 x = 2.060660172 Simplifying x = 2.060660172Subproblem 2
x + -1 = -1.060660172 Simplifying x + -1 = -1.060660172 Reorder the terms: -1 + x = -1.060660172 Solving -1 + x = -1.060660172 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 = -1.060660172 + 1 Combine like terms: -1 + 1 = 0 0 + x = -1.060660172 + 1 x = -1.060660172 + 1 Combine like terms: -1.060660172 + 1 = -0.060660172 x = -0.060660172 Simplifying x = -0.060660172Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.060660172, -0.060660172}Solution
x = {2.060660172, -0.060660172}
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