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Simplifying (5x + 2)(x + 1) = 1 Reorder the terms: (2 + 5x)(x + 1) = 1 Reorder the terms: (2 + 5x)(1 + x) = 1 Multiply (2 + 5x) * (1 + x) (2(1 + x) + 5x * (1 + x)) = 1 ((1 * 2 + x * 2) + 5x * (1 + x)) = 1 ((2 + 2x) + 5x * (1 + x)) = 1 (2 + 2x + (1 * 5x + x * 5x)) = 1 (2 + 2x + (5x + 5x2)) = 1 Combine like terms: 2x + 5x = 7x (2 + 7x + 5x2) = 1 Solving 2 + 7x + 5x2 = 1 Solving for variable 'x'. Reorder the terms: 2 + -1 + 7x + 5x2 = 1 + -1 Combine like terms: 2 + -1 = 1 1 + 7x + 5x2 = 1 + -1 Combine like terms: 1 + -1 = 0 1 + 7x + 5x2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. 0.2 + 1.4x + x2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + 1.4x + -0.2 + x2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + 1.4x + x2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + 1.4x + x2 = 0 + -0.2 1.4x + x2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 1.4x + x2 = -0.2 The x term is 1.4x. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4x + 0.49 + x2 = -0.2 + 0.49 Reorder the terms: 0.49 + 1.4x + x2 = -0.2 + 0.49 Combine like terms: -0.2 + 0.49 = 0.29 0.49 + 1.4x + x2 = 0.29 Factor a perfect square on the left side: (x + 0.7)(x + 0.7) = 0.29 Calculate the square root of the right side: 0.538516481 Break this problem into two subproblems by setting (x + 0.7) equal to 0.538516481 and -0.538516481.Subproblem 1
x + 0.7 = 0.538516481 Simplifying x + 0.7 = 0.538516481 Reorder the terms: 0.7 + x = 0.538516481 Solving 0.7 + x = 0.538516481 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + x = 0.538516481 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + x = 0.538516481 + -0.7 x = 0.538516481 + -0.7 Combine like terms: 0.538516481 + -0.7 = -0.161483519 x = -0.161483519 Simplifying x = -0.161483519Subproblem 2
x + 0.7 = -0.538516481 Simplifying x + 0.7 = -0.538516481 Reorder the terms: 0.7 + x = -0.538516481 Solving 0.7 + x = -0.538516481 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + x = -0.538516481 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + x = -0.538516481 + -0.7 x = -0.538516481 + -0.7 Combine like terms: -0.538516481 + -0.7 = -1.238516481 x = -1.238516481 Simplifying x = -1.238516481Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.161483519, -1.238516481}
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