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Simplifying (2x + 1)(x) + -150 = 0 Reorder the terms: (1 + 2x)(x) + -150 = 0 Reorder the terms for easier multiplication: x(1 + 2x) + -150 = 0 (1 * x + 2x * x) + -150 = 0 (1x + 2x2) + -150 = 0 Reorder the terms: -150 + 1x + 2x2 = 0 Solving -150 + 1x + 2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -75 + 0.5x + x2 = 0 Move the constant term to the right: Add '75' to each side of the equation. -75 + 0.5x + 75 + x2 = 0 + 75 Reorder the terms: -75 + 75 + 0.5x + x2 = 0 + 75 Combine like terms: -75 + 75 = 0 0 + 0.5x + x2 = 0 + 75 0.5x + x2 = 0 + 75 Combine like terms: 0 + 75 = 75 0.5x + x2 = 75 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 = 75 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 75 + 0.0625 Combine like terms: 75 + 0.0625 = 75.0625 0.0625 + 0.5x + x2 = 75.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 75.0625 Calculate the square root of the right side: 8.663861726 Break this problem into two subproblems by setting (x + 0.25) equal to 8.663861726 and -8.663861726.Subproblem 1
x + 0.25 = 8.663861726 Simplifying x + 0.25 = 8.663861726 Reorder the terms: 0.25 + x = 8.663861726 Solving 0.25 + x = 8.663861726 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 = 8.663861726 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 8.663861726 + -0.25 x = 8.663861726 + -0.25 Combine like terms: 8.663861726 + -0.25 = 8.413861726 x = 8.413861726 Simplifying x = 8.413861726Subproblem 2
x + 0.25 = -8.663861726 Simplifying x + 0.25 = -8.663861726 Reorder the terms: 0.25 + x = -8.663861726 Solving 0.25 + x = -8.663861726 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 = -8.663861726 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -8.663861726 + -0.25 x = -8.663861726 + -0.25 Combine like terms: -8.663861726 + -0.25 = -8.913861726 x = -8.913861726 Simplifying x = -8.913861726Solution
The solution to the problem is based on the solutions from the subproblems. x = {8.413861726, -8.913861726}
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