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Simplifying -2.5x + 1 = x2 + -1x Reorder the terms: 1 + -2.5x = x2 + -1x Reorder the terms: 1 + -2.5x = -1x + x2 Solving 1 + -2.5x = -1x + x2 Solving for variable 'x'. Combine like terms: -2.5x + x = -1.5x 1 + -1.5x + -1x2 = -1x + x2 + x + -1x2 Reorder the terms: 1 + -1.5x + -1x2 = -1x + x + x2 + -1x2 Combine like terms: -1x + x = 0 1 + -1.5x + -1x2 = 0 + x2 + -1x2 1 + -1.5x + -1x2 = x2 + -1x2 Combine like terms: x2 + -1x2 = 0 1 + -1.5x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + 1.5x + x2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 1.5x + 1 + x2 = 0 + 1 Reorder the terms: -1 + 1 + 1.5x + x2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 1.5x + x2 = 0 + 1 1.5x + x2 = 0 + 1 Combine like terms: 0 + 1 = 1 1.5x + x2 = 1 The x term is 1.5x. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5x + 0.5625 + x2 = 1 + 0.5625 Reorder the terms: 0.5625 + 1.5x + x2 = 1 + 0.5625 Combine like terms: 1 + 0.5625 = 1.5625 0.5625 + 1.5x + x2 = 1.5625 Factor a perfect square on the left side: (x + 0.75)(x + 0.75) = 1.5625 Calculate the square root of the right side: 1.25 Break this problem into two subproblems by setting (x + 0.75) equal to 1.25 and -1.25.Subproblem 1
x + 0.75 = 1.25 Simplifying x + 0.75 = 1.25 Reorder the terms: 0.75 + x = 1.25 Solving 0.75 + x = 1.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = 1.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = 1.25 + -0.75 x = 1.25 + -0.75 Combine like terms: 1.25 + -0.75 = 0.5 x = 0.5 Simplifying x = 0.5Subproblem 2
x + 0.75 = -1.25 Simplifying x + 0.75 = -1.25 Reorder the terms: 0.75 + x = -1.25 Solving 0.75 + x = -1.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + x = -1.25 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + x = -1.25 + -0.75 x = -1.25 + -0.75 Combine like terms: -1.25 + -0.75 = -2 x = -2 Simplifying x = -2Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.5, -2}
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