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Simplifying (2x + 1) * x = 274 Reorder the terms: (1 + 2x) * x = 274 Reorder the terms for easier multiplication: x(1 + 2x) = 274 (1 * x + 2x * x) = 274 (1x + 2x2) = 274 Solving 1x + 2x2 = 274 Solving for variable 'x'. Reorder the terms: -274 + 1x + 2x2 = 274 + -274 Combine like terms: 274 + -274 = 0 -274 + 1x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -137 + 0.5x + x2 = 0 Move the constant term to the right: Add '137' to each side of the equation. -137 + 0.5x + 137 + x2 = 0 + 137 Reorder the terms: -137 + 137 + 0.5x + x2 = 0 + 137 Combine like terms: -137 + 137 = 0 0 + 0.5x + x2 = 0 + 137 0.5x + x2 = 0 + 137 Combine like terms: 0 + 137 = 137 0.5x + x2 = 137 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = 137 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 137 + 0.0625 Combine like terms: 137 + 0.0625 = 137.0625 0.0625 + 0.5x + x2 = 137.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 137.0625 Calculate the square root of the right side: 11.707369474 Break this problem into two subproblems by setting (x + 0.25) equal to 11.707369474 and -11.707369474.Subproblem 1
x + 0.25 = 11.707369474 Simplifying x + 0.25 = 11.707369474 Reorder the terms: 0.25 + x = 11.707369474 Solving 0.25 + x = 11.707369474 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 11.707369474 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 11.707369474 + -0.25 x = 11.707369474 + -0.25 Combine like terms: 11.707369474 + -0.25 = 11.457369474 x = 11.457369474 Simplifying x = 11.457369474Subproblem 2
x + 0.25 = -11.707369474 Simplifying x + 0.25 = -11.707369474 Reorder the terms: 0.25 + x = -11.707369474 Solving 0.25 + x = -11.707369474 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -11.707369474 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -11.707369474 + -0.25 x = -11.707369474 + -0.25 Combine like terms: -11.707369474 + -0.25 = -11.957369474 x = -11.957369474 Simplifying x = -11.957369474Solution
The solution to the problem is based on the solutions from the subproblems. x = {11.457369474, -11.957369474}
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