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Simplifying 0.25x2 + 0.75x = 1 Reorder the terms: 0.75x + 0.25x2 = 1 Solving 0.75x + 0.25x2 = 1 Solving for variable 'x'. Reorder the terms: -1 + 0.75x + 0.25x2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 0.75x + 0.25x2 = 0 Begin completing the square. Divide all terms by 0.25 the coefficient of the squared term: Divide each side by '0.25'. -4 + 3x + x2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 3x + 4 + x2 = 0 + 4 Reorder the terms: -4 + 4 + 3x + x2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 3x + x2 = 0 + 4 3x + x2 = 0 + 4 Combine like terms: 0 + 4 = 4 3x + x2 = 4 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 4 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 4 + 2.25 Combine like terms: 4 + 2.25 = 6.25 2.25 + 3x + x2 = 6.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 6.25 Calculate the square root of the right side: 2.5 Break this problem into two subproblems by setting (x + 1.5) equal to 2.5 and -2.5.Subproblem 1
x + 1.5 = 2.5 Simplifying x + 1.5 = 2.5 Reorder the terms: 1.5 + x = 2.5 Solving 1.5 + x = 2.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 2.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 2.5 + -1.5 x = 2.5 + -1.5 Combine like terms: 2.5 + -1.5 = 1 x = 1 Simplifying x = 1Subproblem 2
x + 1.5 = -2.5 Simplifying x + 1.5 = -2.5 Reorder the terms: 1.5 + x = -2.5 Solving 1.5 + x = -2.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -2.5 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -2.5 + -1.5 x = -2.5 + -1.5 Combine like terms: -2.5 + -1.5 = -4 x = -4 Simplifying x = -4Solution
The solution to the problem is based on the solutions from the subproblems. x = {1, -4}
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