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Simplifying (5x + 2)(2x + 1) = 27 Reorder the terms: (2 + 5x)(2x + 1) = 27 Reorder the terms: (2 + 5x)(1 + 2x) = 27 Multiply (2 + 5x) * (1 + 2x) (2(1 + 2x) + 5x * (1 + 2x)) = 27 ((1 * 2 + 2x * 2) + 5x * (1 + 2x)) = 27 ((2 + 4x) + 5x * (1 + 2x)) = 27 (2 + 4x + (1 * 5x + 2x * 5x)) = 27 (2 + 4x + (5x + 10x2)) = 27 Combine like terms: 4x + 5x = 9x (2 + 9x + 10x2) = 27 Solving 2 + 9x + 10x2 = 27 Solving for variable 'x'. Reorder the terms: 2 + -27 + 9x + 10x2 = 27 + -27 Combine like terms: 2 + -27 = -25 -25 + 9x + 10x2 = 27 + -27 Combine like terms: 27 + -27 = 0 -25 + 9x + 10x2 = 0 Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -2.5 + 0.9x + x2 = 0.0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 0.9x + 2.5 + x2 = 0.0 + 2.5 Reorder the terms: -2.5 + 2.5 + 0.9x + x2 = 0.0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 0.9x + x2 = 0.0 + 2.5 0.9x + x2 = 0.0 + 2.5 Combine like terms: 0.0 + 2.5 = 2.5 0.9x + x2 = 2.5 The x term is 0.9x. Take half its coefficient (0.45). Square it (0.2025) and add it to both sides. Add '0.2025' to each side of the equation. 0.9x + 0.2025 + x2 = 2.5 + 0.2025 Reorder the terms: 0.2025 + 0.9x + x2 = 2.5 + 0.2025 Combine like terms: 2.5 + 0.2025 = 2.7025 0.2025 + 0.9x + x2 = 2.7025 Factor a perfect square on the left side: (x + 0.45)(x + 0.45) = 2.7025 Calculate the square root of the right side: 1.643928222 Break this problem into two subproblems by setting (x + 0.45) equal to 1.643928222 and -1.643928222.Subproblem 1
x + 0.45 = 1.643928222 Simplifying x + 0.45 = 1.643928222 Reorder the terms: 0.45 + x = 1.643928222 Solving 0.45 + x = 1.643928222 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + x = 1.643928222 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + x = 1.643928222 + -0.45 x = 1.643928222 + -0.45 Combine like terms: 1.643928222 + -0.45 = 1.193928222 x = 1.193928222 Simplifying x = 1.193928222Subproblem 2
x + 0.45 = -1.643928222 Simplifying x + 0.45 = -1.643928222 Reorder the terms: 0.45 + x = -1.643928222 Solving 0.45 + x = -1.643928222 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.45' to each side of the equation. 0.45 + -0.45 + x = -1.643928222 + -0.45 Combine like terms: 0.45 + -0.45 = 0.00 0.00 + x = -1.643928222 + -0.45 x = -1.643928222 + -0.45 Combine like terms: -1.643928222 + -0.45 = -2.093928222 x = -2.093928222 Simplifying x = -2.093928222Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.193928222, -2.093928222}
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