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Simplifying (4x + -5)(x + 2) = 1 Reorder the terms: (-5 + 4x)(x + 2) = 1 Reorder the terms: (-5 + 4x)(2 + x) = 1 Multiply (-5 + 4x) * (2 + x) (-5(2 + x) + 4x * (2 + x)) = 1 ((2 * -5 + x * -5) + 4x * (2 + x)) = 1 ((-10 + -5x) + 4x * (2 + x)) = 1 (-10 + -5x + (2 * 4x + x * 4x)) = 1 (-10 + -5x + (8x + 4x2)) = 1 Combine like terms: -5x + 8x = 3x (-10 + 3x + 4x2) = 1 Solving -10 + 3x + 4x2 = 1 Solving for variable 'x'. Reorder the terms: -10 + -1 + 3x + 4x2 = 1 + -1 Combine like terms: -10 + -1 = -11 -11 + 3x + 4x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -11 + 3x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -2.75 + 0.75x + x2 = 0 Move the constant term to the right: Add '2.75' to each side of the equation. -2.75 + 0.75x + 2.75 + x2 = 0 + 2.75 Reorder the terms: -2.75 + 2.75 + 0.75x + x2 = 0 + 2.75 Combine like terms: -2.75 + 2.75 = 0.00 0.00 + 0.75x + x2 = 0 + 2.75 0.75x + x2 = 0 + 2.75 Combine like terms: 0 + 2.75 = 2.75 0.75x + x2 = 2.75 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 + x2 = 2.75 + 0.140625 Reorder the terms: 0.140625 + 0.75x + x2 = 2.75 + 0.140625 Combine like terms: 2.75 + 0.140625 = 2.890625 0.140625 + 0.75x + x2 = 2.890625 Factor a perfect square on the left side: (x + 0.375)(x + 0.375) = 2.890625 Calculate the square root of the right side: 1.700183814 Break this problem into two subproblems by setting (x + 0.375) equal to 1.700183814 and -1.700183814.Subproblem 1
x + 0.375 = 1.700183814 Simplifying x + 0.375 = 1.700183814 Reorder the terms: 0.375 + x = 1.700183814 Solving 0.375 + x = 1.700183814 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 = 1.700183814 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 1.700183814 + -0.375 x = 1.700183814 + -0.375 Combine like terms: 1.700183814 + -0.375 = 1.325183814 x = 1.325183814 Simplifying x = 1.325183814Subproblem 2
x + 0.375 = -1.700183814 Simplifying x + 0.375 = -1.700183814 Reorder the terms: 0.375 + x = -1.700183814 Solving 0.375 + x = -1.700183814 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 = -1.700183814 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -1.700183814 + -0.375 x = -1.700183814 + -0.375 Combine like terms: -1.700183814 + -0.375 = -2.075183814 x = -2.075183814 Simplifying x = -2.075183814Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.325183814, -2.075183814}
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