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Simplifying (2x + 1)(4x + -1) = 32 Reorder the terms: (1 + 2x)(4x + -1) = 32 Reorder the terms: (1 + 2x)(-1 + 4x) = 32 Multiply (1 + 2x) * (-1 + 4x) (1(-1 + 4x) + 2x * (-1 + 4x)) = 32 ((-1 * 1 + 4x * 1) + 2x * (-1 + 4x)) = 32 ((-1 + 4x) + 2x * (-1 + 4x)) = 32 (-1 + 4x + (-1 * 2x + 4x * 2x)) = 32 (-1 + 4x + (-2x + 8x2)) = 32 Combine like terms: 4x + -2x = 2x (-1 + 2x + 8x2) = 32 Solving -1 + 2x + 8x2 = 32 Solving for variable 'x'. Reorder the terms: -1 + -32 + 2x + 8x2 = 32 + -32 Combine like terms: -1 + -32 = -33 -33 + 2x + 8x2 = 32 + -32 Combine like terms: 32 + -32 = 0 -33 + 2x + 8x2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -4.125 + 0.25x + x2 = 0 Move the constant term to the right: Add '4.125' to each side of the equation. -4.125 + 0.25x + 4.125 + x2 = 0 + 4.125 Reorder the terms: -4.125 + 4.125 + 0.25x + x2 = 0 + 4.125 Combine like terms: -4.125 + 4.125 = 0.000 0.000 + 0.25x + x2 = 0 + 4.125 0.25x + x2 = 0 + 4.125 Combine like terms: 0 + 4.125 = 4.125 0.25x + x2 = 4.125 The x term is 0.25x. Take half its coefficient (0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. 0.25x + 0.015625 + x2 = 4.125 + 0.015625 Reorder the terms: 0.015625 + 0.25x + x2 = 4.125 + 0.015625 Combine like terms: 4.125 + 0.015625 = 4.140625 0.015625 + 0.25x + x2 = 4.140625 Factor a perfect square on the left side: (x + 0.125)(x + 0.125) = 4.140625 Calculate the square root of the right side: 2.034852575 Break this problem into two subproblems by setting (x + 0.125) equal to 2.034852575 and -2.034852575.Subproblem 1
x + 0.125 = 2.034852575 Simplifying x + 0.125 = 2.034852575 Reorder the terms: 0.125 + x = 2.034852575 Solving 0.125 + x = 2.034852575 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = 2.034852575 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = 2.034852575 + -0.125 x = 2.034852575 + -0.125 Combine like terms: 2.034852575 + -0.125 = 1.909852575 x = 1.909852575 Simplifying x = 1.909852575Subproblem 2
x + 0.125 = -2.034852575 Simplifying x + 0.125 = -2.034852575 Reorder the terms: 0.125 + x = -2.034852575 Solving 0.125 + x = -2.034852575 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.125' to each side of the equation. 0.125 + -0.125 + x = -2.034852575 + -0.125 Combine like terms: 0.125 + -0.125 = 0.000 0.000 + x = -2.034852575 + -0.125 x = -2.034852575 + -0.125 Combine like terms: -2.034852575 + -0.125 = -2.159852575 x = -2.159852575 Simplifying x = -2.159852575Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.909852575, -2.159852575}
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