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Simplifying (4x + 5)(5x + 7) = 7 Reorder the terms: (5 + 4x)(5x + 7) = 7 Reorder the terms: (5 + 4x)(7 + 5x) = 7 Multiply (5 + 4x) * (7 + 5x) (5(7 + 5x) + 4x * (7 + 5x)) = 7 ((7 * 5 + 5x * 5) + 4x * (7 + 5x)) = 7 ((35 + 25x) + 4x * (7 + 5x)) = 7 (35 + 25x + (7 * 4x + 5x * 4x)) = 7 (35 + 25x + (28x + 20x2)) = 7 Combine like terms: 25x + 28x = 53x (35 + 53x + 20x2) = 7 Solving 35 + 53x + 20x2 = 7 Solving for variable 'x'. Reorder the terms: 35 + -7 + 53x + 20x2 = 7 + -7 Combine like terms: 35 + -7 = 28 28 + 53x + 20x2 = 7 + -7 Combine like terms: 7 + -7 = 0 28 + 53x + 20x2 = 0 Begin completing the square. Divide all terms by 20 the coefficient of the squared term: Divide each side by '20'. 1.4 + 2.65x + x2 = 0.0 Move the constant term to the right: Add '-1.4' to each side of the equation. 1.4 + 2.65x + -1.4 + x2 = 0.0 + -1.4 Reorder the terms: 1.4 + -1.4 + 2.65x + x2 = 0.0 + -1.4 Combine like terms: 1.4 + -1.4 = 0.0 0.0 + 2.65x + x2 = 0.0 + -1.4 2.65x + x2 = 0.0 + -1.4 Combine like terms: 0.0 + -1.4 = -1.4 2.65x + x2 = -1.4 The x term is 2.65x. Take half its coefficient (1.325). Square it (1.755625) and add it to both sides. Add '1.755625' to each side of the equation. 2.65x + 1.755625 + x2 = -1.4 + 1.755625 Reorder the terms: 1.755625 + 2.65x + x2 = -1.4 + 1.755625 Combine like terms: -1.4 + 1.755625 = 0.355625 1.755625 + 2.65x + x2 = 0.355625 Factor a perfect square on the left side: (x + 1.325)(x + 1.325) = 0.355625 Calculate the square root of the right side: 0.596343022 Break this problem into two subproblems by setting (x + 1.325) equal to 0.596343022 and -0.596343022.Subproblem 1
x + 1.325 = 0.596343022 Simplifying x + 1.325 = 0.596343022 Reorder the terms: 1.325 + x = 0.596343022 Solving 1.325 + x = 0.596343022 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.325' to each side of the equation. 1.325 + -1.325 + x = 0.596343022 + -1.325 Combine like terms: 1.325 + -1.325 = 0.000 0.000 + x = 0.596343022 + -1.325 x = 0.596343022 + -1.325 Combine like terms: 0.596343022 + -1.325 = -0.728656978 x = -0.728656978 Simplifying x = -0.728656978Subproblem 2
x + 1.325 = -0.596343022 Simplifying x + 1.325 = -0.596343022 Reorder the terms: 1.325 + x = -0.596343022 Solving 1.325 + x = -0.596343022 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.325' to each side of the equation. 1.325 + -1.325 + x = -0.596343022 + -1.325 Combine like terms: 1.325 + -1.325 = 0.000 0.000 + x = -0.596343022 + -1.325 x = -0.596343022 + -1.325 Combine like terms: -0.596343022 + -1.325 = -1.921343022 x = -1.921343022 Simplifying x = -1.921343022Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.728656978, -1.921343022}
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