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Simplifying 7x + 5 = (2x + 3)(4x + 1) Reorder the terms: 5 + 7x = (2x + 3)(4x + 1) Reorder the terms: 5 + 7x = (3 + 2x)(4x + 1) Reorder the terms: 5 + 7x = (3 + 2x)(1 + 4x) Multiply (3 + 2x) * (1 + 4x) 5 + 7x = (3(1 + 4x) + 2x * (1 + 4x)) 5 + 7x = ((1 * 3 + 4x * 3) + 2x * (1 + 4x)) 5 + 7x = ((3 + 12x) + 2x * (1 + 4x)) 5 + 7x = (3 + 12x + (1 * 2x + 4x * 2x)) 5 + 7x = (3 + 12x + (2x + 8x2)) Combine like terms: 12x + 2x = 14x 5 + 7x = (3 + 14x + 8x2) Solving 5 + 7x = 3 + 14x + 8x2 Solving for variable 'x'. Reorder the terms: 5 + -3 + 7x + -14x + -8x2 = 3 + 14x + 8x2 + -3 + -14x + -8x2 Combine like terms: 5 + -3 = 2 2 + 7x + -14x + -8x2 = 3 + 14x + 8x2 + -3 + -14x + -8x2 Combine like terms: 7x + -14x = -7x 2 + -7x + -8x2 = 3 + 14x + 8x2 + -3 + -14x + -8x2 Reorder the terms: 2 + -7x + -8x2 = 3 + -3 + 14x + -14x + 8x2 + -8x2 Combine like terms: 3 + -3 = 0 2 + -7x + -8x2 = 0 + 14x + -14x + 8x2 + -8x2 2 + -7x + -8x2 = 14x + -14x + 8x2 + -8x2 Combine like terms: 14x + -14x = 0 2 + -7x + -8x2 = 0 + 8x2 + -8x2 2 + -7x + -8x2 = 8x2 + -8x2 Combine like terms: 8x2 + -8x2 = 0 2 + -7x + -8x2 = 0 Begin completing the square. Divide all terms by -8 the coefficient of the squared term: Divide each side by '-8'. -0.25 + 0.875x + x2 = 0 Move the constant term to the right: Add '0.25' to each side of the equation. -0.25 + 0.875x + 0.25 + x2 = 0 + 0.25 Reorder the terms: -0.25 + 0.25 + 0.875x + x2 = 0 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + 0.875x + x2 = 0 + 0.25 0.875x + x2 = 0 + 0.25 Combine like terms: 0 + 0.25 = 0.25 0.875x + x2 = 0.25 The x term is 0.875x. Take half its coefficient (0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. 0.875x + 0.19140625 + x2 = 0.25 + 0.19140625 Reorder the terms: 0.19140625 + 0.875x + x2 = 0.25 + 0.19140625 Combine like terms: 0.25 + 0.19140625 = 0.44140625 0.19140625 + 0.875x + x2 = 0.44140625 Factor a perfect square on the left side: (x + 0.4375)(x + 0.4375) = 0.44140625 Calculate the square root of the right side: 0.664384113 Break this problem into two subproblems by setting (x + 0.4375) equal to 0.664384113 and -0.664384113.Subproblem 1
x + 0.4375 = 0.664384113 Simplifying x + 0.4375 = 0.664384113 Reorder the terms: 0.4375 + x = 0.664384113 Solving 0.4375 + x = 0.664384113 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + x = 0.664384113 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + x = 0.664384113 + -0.4375 x = 0.664384113 + -0.4375 Combine like terms: 0.664384113 + -0.4375 = 0.226884113 x = 0.226884113 Simplifying x = 0.226884113Subproblem 2
x + 0.4375 = -0.664384113 Simplifying x + 0.4375 = -0.664384113 Reorder the terms: 0.4375 + x = -0.664384113 Solving 0.4375 + x = -0.664384113 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + x = -0.664384113 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + x = -0.664384113 + -0.4375 x = -0.664384113 + -0.4375 Combine like terms: -0.664384113 + -0.4375 = -1.101884113 x = -1.101884113 Simplifying x = -1.101884113Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.226884113, -1.101884113}
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