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Simplifying (2x + -1)(4x + -1) = 14 Reorder the terms: (-1 + 2x)(4x + -1) = 14 Reorder the terms: (-1 + 2x)(-1 + 4x) = 14 Multiply (-1 + 2x) * (-1 + 4x) (-1(-1 + 4x) + 2x * (-1 + 4x)) = 14 ((-1 * -1 + 4x * -1) + 2x * (-1 + 4x)) = 14 ((1 + -4x) + 2x * (-1 + 4x)) = 14 (1 + -4x + (-1 * 2x + 4x * 2x)) = 14 (1 + -4x + (-2x + 8x^{2})) = 14 Combine like terms: -4x + -2x = -6x (1 + -6x + 8x^{2}) = 14 Solving 1 + -6x + 8x^{2}= 14 Solving for variable 'x'. Reorder the terms: 1 + -14 + -6x + 8x^{2}= 14 + -14 Combine like terms: 1 + -14 = -13 -13 + -6x + 8x^{2}= 14 + -14 Combine like terms: 14 + -14 = 0 -13 + -6x + 8x^{2}= 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -1.625 + -0.75x + x^{2}= 0 Move the constant term to the right: Add '1.625' to each side of the equation. -1.625 + -0.75x + 1.625 + x^{2}= 0 + 1.625 Reorder the terms: -1.625 + 1.625 + -0.75x + x^{2}= 0 + 1.625 Combine like terms: -1.625 + 1.625 = 0.000 0.000 + -0.75x + x^{2}= 0 + 1.625 -0.75x + x^{2}= 0 + 1.625 Combine like terms: 0 + 1.625 = 1.625 -0.75x + x^{2}= 1.625 The x term is -0.75x. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75x + 0.140625 + x^{2}= 1.625 + 0.140625 Reorder the terms: 0.140625 + -0.75x + x^{2}= 1.625 + 0.140625 Combine like terms: 1.625 + 0.140625 = 1.765625 0.140625 + -0.75x + x^{2}= 1.765625 Factor a perfect square on the left side: (x + -0.375)(x + -0.375) = 1.765625 Calculate the square root of the right side: 1.328768227 Break this problem into two subproblems by setting (x + -0.375) equal to 1.328768227 and -1.328768227.## Subproblem 1

x + -0.375 = 1.328768227 Simplifying x + -0.375 = 1.328768227 Reorder the terms: -0.375 + x = 1.328768227 Solving -0.375 + x = 1.328768227 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 1.328768227 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 1.328768227 + 0.375 x = 1.328768227 + 0.375 Combine like terms: 1.328768227 + 0.375 = 1.703768227 x = 1.703768227 Simplifying x = 1.703768227## Subproblem 2

x + -0.375 = -1.328768227 Simplifying x + -0.375 = -1.328768227 Reorder the terms: -0.375 + x = -1.328768227 Solving -0.375 + x = -1.328768227 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -1.328768227 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -1.328768227 + 0.375 x = -1.328768227 + 0.375 Combine like terms: -1.328768227 + 0.375 = -0.953768227 x = -0.953768227 Simplifying x = -0.953768227## Solution

The solution to the problem is based on the solutions from the subproblems. x = {1.703768227, -0.953768227}

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