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Simplifying 2x(x + -1) = 1 + -1(2 + -3x) Reorder the terms: 2x(-1 + x) = 1 + -1(2 + -3x) (-1 * 2x + x * 2x) = 1 + -1(2 + -3x) (-2x + 2x2) = 1 + -1(2 + -3x) -2x + 2x2 = 1 + (2 * -1 + -3x * -1) -2x + 2x2 = 1 + (-2 + 3x) Combine like terms: 1 + -2 = -1 -2x + 2x2 = -1 + 3x Solving -2x + 2x2 = -1 + 3x Solving for variable 'x'. Reorder the terms: 1 + -2x + -3x + 2x2 = -1 + 3x + 1 + -3x Combine like terms: -2x + -3x = -5x 1 + -5x + 2x2 = -1 + 3x + 1 + -3x Reorder the terms: 1 + -5x + 2x2 = -1 + 1 + 3x + -3x Combine like terms: -1 + 1 = 0 1 + -5x + 2x2 = 0 + 3x + -3x 1 + -5x + 2x2 = 3x + -3x Combine like terms: 3x + -3x = 0 1 + -5x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 0.5 + -2.5x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2.5x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2.5x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2.5x + x2 = 0 + -0.5 -2.5x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2.5x + x2 = -0.5 The x term is -2.5x. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5x + 1.5625 + x2 = -0.5 + 1.5625 Reorder the terms: 1.5625 + -2.5x + x2 = -0.5 + 1.5625 Combine like terms: -0.5 + 1.5625 = 1.0625 1.5625 + -2.5x + x2 = 1.0625 Factor a perfect square on the left side: (x + -1.25)(x + -1.25) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting (x + -1.25) equal to 1.030776406 and -1.030776406.Subproblem 1
x + -1.25 = 1.030776406 Simplifying x + -1.25 = 1.030776406 Reorder the terms: -1.25 + x = 1.030776406 Solving -1.25 + x = 1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = 1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = 1.030776406 + 1.25 x = 1.030776406 + 1.25 Combine like terms: 1.030776406 + 1.25 = 2.280776406 x = 2.280776406 Simplifying x = 2.280776406Subproblem 2
x + -1.25 = -1.030776406 Simplifying x + -1.25 = -1.030776406 Reorder the terms: -1.25 + x = -1.030776406 Solving -1.25 + x = -1.030776406 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + x = -1.030776406 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + x = -1.030776406 + 1.25 x = -1.030776406 + 1.25 Combine like terms: -1.030776406 + 1.25 = 0.219223594 x = 0.219223594 Simplifying x = 0.219223594Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.280776406, 0.219223594}
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