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Simplifying -1x(2x + 1) = -24 Reorder the terms: -1x(1 + 2x) = -24 (1 * -1x + 2x * -1x) = -24 (-1x + -2x2) = -24 Solving -1x + -2x2 = -24 Solving for variable 'x'. Reorder the terms: 24 + -1x + -2x2 = -24 + 24 Combine like terms: -24 + 24 = 0 24 + -1x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -12 + 0.5x + x2 = 0 Move the constant term to the right: Add '12' to each side of the equation. -12 + 0.5x + 12 + x2 = 0 + 12 Reorder the terms: -12 + 12 + 0.5x + x2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 0.5x + x2 = 0 + 12 0.5x + x2 = 0 + 12 Combine like terms: 0 + 12 = 12 0.5x + x2 = 12 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 = 12 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = 12 + 0.0625 Combine like terms: 12 + 0.0625 = 12.0625 0.0625 + 0.5x + x2 = 12.0625 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 12.0625 Calculate the square root of the right side: 3.473110997 Break this problem into two subproblems by setting (x + 0.25) equal to 3.473110997 and -3.473110997.Subproblem 1
x + 0.25 = 3.473110997 Simplifying x + 0.25 = 3.473110997 Reorder the terms: 0.25 + x = 3.473110997 Solving 0.25 + x = 3.473110997 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 = 3.473110997 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 3.473110997 + -0.25 x = 3.473110997 + -0.25 Combine like terms: 3.473110997 + -0.25 = 3.223110997 x = 3.223110997 Simplifying x = 3.223110997Subproblem 2
x + 0.25 = -3.473110997 Simplifying x + 0.25 = -3.473110997 Reorder the terms: 0.25 + x = -3.473110997 Solving 0.25 + x = -3.473110997 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 = -3.473110997 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -3.473110997 + -0.25 x = -3.473110997 + -0.25 Combine like terms: -3.473110997 + -0.25 = -3.723110997 x = -3.723110997 Simplifying x = -3.723110997Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.223110997, -3.723110997}
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