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Simplifying -2x2 + x + 24 = 0 Reorder the terms: 24 + x + -2x2 = 0 Solving 24 + x + -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'. -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 x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.5x + 0.25 + x2 = 12 + 0.25 Reorder the terms: 0.25 + -0.5x + x2 = 12 + 0.25 Combine like terms: 12 + 0.25 = 12.25 0.25 + -0.5x + x2 = 12.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 12.25 Calculate the square root of the right side: 3.5 Break this problem into two subproblems by setting (x + 0.5) equal to 3.5 and -3.5.Subproblem 1
x + 0.5 = 3.5 Simplifying x + 0.5 = 3.5 Reorder the terms: 0.5 + x = 3.5 Solving 0.5 + x = 3.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 3.5 + -0.5 x = 3.5 + -0.5 Combine like terms: 3.5 + -0.5 = 3 x = 3 Simplifying x = 3Subproblem 2
x + 0.5 = -3.5 Simplifying x + 0.5 = -3.5 Reorder the terms: 0.5 + x = -3.5 Solving 0.5 + x = -3.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -3.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -3.5 + -0.5 x = -3.5 + -0.5 Combine like terms: -3.5 + -0.5 = -4 x = -4 Simplifying x = -4Solution
The solution to the problem is based on the solutions from the subproblems. x = {3, -4}
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