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Simplifying 150 = (0.5)(2x + 1)(x) Reorder the terms: 150 = 0.5(1 + 2x)(x) Reorder the terms for easier multiplication: 150 = 0.5x(1 + 2x) 150 = (1 * 0.5x + 2x * 0.5x) 150 = (0.5x + 1x2) Solving 150 = 0.5x + 1x2 Solving for variable 'x'. Reorder the terms: 150 + -0.5x + -1x2 = 0.5x + -0.5x + 1x2 + -1x2 Combine like terms: 0.5x + -0.5x = 0.0 150 + -0.5x + -1x2 = 0.0 + 1x2 + -1x2 150 + -0.5x + -1x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 150 + -0.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -150 + 0.5x + x2 = 0 Move the constant term to the right: Add '150' to each side of the equation. -150 + 0.5x + 150 + x2 = 0 + 150 Reorder the terms: -150 + 150 + 0.5x + x2 = 0 + 150 Combine like terms: -150 + 150 = 0 0 + 0.5x + x2 = 0 + 150 0.5x + x2 = 0 + 150 Combine like terms: 0 + 150 = 150 0.5x + x2 = 150 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 = 150 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 150 + 0.0625 Combine like terms: 150 + 0.0625 = 150.0625 0.0625 + 0.5x + x2 = 150.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 150.0625 Calculate the square root of the right side: 12.25 Break this problem into two subproblems by setting (x + 0.25) equal to 12.25 and -12.25.Subproblem 1
x + 0.25 = 12.25 Simplifying x + 0.25 = 12.25 Reorder the terms: 0.25 + x = 12.25 Solving 0.25 + x = 12.25 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 = 12.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 12.25 + -0.25 x = 12.25 + -0.25 Combine like terms: 12.25 + -0.25 = 12 x = 12 Simplifying x = 12Subproblem 2
x + 0.25 = -12.25 Simplifying x + 0.25 = -12.25 Reorder the terms: 0.25 + x = -12.25 Solving 0.25 + x = -12.25 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 = -12.25 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -12.25 + -0.25 x = -12.25 + -0.25 Combine like terms: -12.25 + -0.25 = -12.5 x = -12.5 Simplifying x = -12.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {12, -12.5}
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