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Simplifying 4x = 2(2x + 1)(2x + -1) Reorder the terms: 4x = 2(1 + 2x)(2x + -1) Reorder the terms: 4x = 2(1 + 2x)(-1 + 2x) Multiply (1 + 2x) * (-1 + 2x) 4x = 2(1(-1 + 2x) + 2x * (-1 + 2x)) 4x = 2((-1 * 1 + 2x * 1) + 2x * (-1 + 2x)) 4x = 2((-1 + 2x) + 2x * (-1 + 2x)) 4x = 2(-1 + 2x + (-1 * 2x + 2x * 2x)) 4x = 2(-1 + 2x + (-2x + 4x2)) Combine like terms: 2x + -2x = 0 4x = 2(-1 + 0 + 4x2) 4x = 2(-1 + 4x2) 4x = (-1 * 2 + 4x2 * 2) 4x = (-2 + 8x2) Solving 4x = -2 + 8x2 Solving for variable 'x'. Reorder the terms: 2 + 4x + -8x2 = -2 + 8x2 + 2 + -8x2 Reorder the terms: 2 + 4x + -8x2 = -2 + 2 + 8x2 + -8x2 Combine like terms: -2 + 2 = 0 2 + 4x + -8x2 = 0 + 8x2 + -8x2 2 + 4x + -8x2 = 8x2 + -8x2 Combine like terms: 8x2 + -8x2 = 0 2 + 4x + -8x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(1 + 2x + -4x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(1 + 2x + -4x2)' equal to zero and attempt to solve: Simplifying 1 + 2x + -4x2 = 0 Solving 1 + 2x + -4x2 = 0 Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -0.25 + -0.5x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + -0.5x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + -0.5x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + -0.5x + x2 = 0 + 0.25 -0.5x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 -0.5x + x2 = 0.25 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 = 0.25 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 0.25 + 0.0625 Combine like terms: 0.25 + 0.0625 = 0.3125 0.0625 + -0.5x + x2 = 0.3125 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 0.3125 Calculate the square root of the right side: 0.559016994 Break this problem into two subproblems by setting (x + -0.25) equal to 0.559016994 and -0.559016994.Subproblem 1
x + -0.25 = 0.559016994 Simplifying x + -0.25 = 0.559016994 Reorder the terms: -0.25 + x = 0.559016994 Solving -0.25 + x = 0.559016994 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 = 0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 0.559016994 + 0.25 x = 0.559016994 + 0.25 Combine like terms: 0.559016994 + 0.25 = 0.809016994 x = 0.809016994 Simplifying x = 0.809016994Subproblem 2
x + -0.25 = -0.559016994 Simplifying x + -0.25 = -0.559016994 Reorder the terms: -0.25 + x = -0.559016994 Solving -0.25 + x = -0.559016994 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 = -0.559016994 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -0.559016994 + 0.25 x = -0.559016994 + 0.25 Combine like terms: -0.559016994 + 0.25 = -0.309016994 x = -0.309016994 Simplifying x = -0.309016994Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.809016994, -0.309016994}Solution
x = {0.809016994, -0.309016994}
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