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Simplifying (4x + -1)(4x + -1) = 11 Reorder the terms: (-1 + 4x)(4x + -1) = 11 Reorder the terms: (-1 + 4x)(-1 + 4x) = 11 Multiply (-1 + 4x) * (-1 + 4x) (-1(-1 + 4x) + 4x * (-1 + 4x)) = 11 ((-1 * -1 + 4x * -1) + 4x * (-1 + 4x)) = 11 ((1 + -4x) + 4x * (-1 + 4x)) = 11 (1 + -4x + (-1 * 4x + 4x * 4x)) = 11 (1 + -4x + (-4x + 16x2)) = 11 Combine like terms: -4x + -4x = -8x (1 + -8x + 16x2) = 11 Solving 1 + -8x + 16x2 = 11 Solving for variable 'x'. Reorder the terms: 1 + -11 + -8x + 16x2 = 11 + -11 Combine like terms: 1 + -11 = -10 -10 + -8x + 16x2 = 11 + -11 Combine like terms: 11 + -11 = 0 -10 + -8x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-5 + -4x + 8x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-5 + -4x + 8x2)' equal to zero and attempt to solve: Simplifying -5 + -4x + 8x2 = 0 Solving -5 + -4x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.625 + -0.5x + x2 = 0 Move the constant term to the right: Add '0.625' to each side of the equation. -0.625 + -0.5x + 0.625 + x2 = 0 + 0.625 Reorder the terms: -0.625 + 0.625 + -0.5x + x2 = 0 + 0.625 Combine like terms: -0.625 + 0.625 = 0.000 0.000 + -0.5x + x2 = 0 + 0.625 -0.5x + x2 = 0 + 0.625 Combine like terms: 0 + 0.625 = 0.625 -0.5x + x2 = 0.625 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 = 0.625 + 0.0625 Reorder the terms: 0.0625 + -0.5x + x2 = 0.625 + 0.0625 Combine like terms: 0.625 + 0.0625 = 0.6875 0.0625 + -0.5x + x2 = 0.6875 Factor a perfect square on the left side: (x + -0.25)(x + -0.25) = 0.6875 Calculate the square root of the right side: 0.829156198 Break this problem into two subproblems by setting (x + -0.25) equal to 0.829156198 and -0.829156198.Subproblem 1
x + -0.25 = 0.829156198 Simplifying x + -0.25 = 0.829156198 Reorder the terms: -0.25 + x = 0.829156198 Solving -0.25 + x = 0.829156198 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 = 0.829156198 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = 0.829156198 + 0.25 x = 0.829156198 + 0.25 Combine like terms: 0.829156198 + 0.25 = 1.079156198 x = 1.079156198 Simplifying x = 1.079156198Subproblem 2
x + -0.25 = -0.829156198 Simplifying x + -0.25 = -0.829156198 Reorder the terms: -0.25 + x = -0.829156198 Solving -0.25 + x = -0.829156198 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 = -0.829156198 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + x = -0.829156198 + 0.25 x = -0.829156198 + 0.25 Combine like terms: -0.829156198 + 0.25 = -0.579156198 x = -0.579156198 Simplifying x = -0.579156198Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.079156198, -0.579156198}Solution
x = {1.079156198, -0.579156198}
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