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Simplifying 5(1 + -1x) + 2x = 2x(1 + x) (1 * 5 + -1x * 5) + 2x = 2x(1 + x) (5 + -5x) + 2x = 2x(1 + x) Combine like terms: -5x + 2x = -3x 5 + -3x = 2x(1 + x) 5 + -3x = (1 * 2x + x * 2x) 5 + -3x = (2x + 2x2) Solving 5 + -3x = 2x + 2x2 Solving for variable 'x'. Combine like terms: -3x + -2x = -5x 5 + -5x + -2x2 = 2x + 2x2 + -2x + -2x2 Reorder the terms: 5 + -5x + -2x2 = 2x + -2x + 2x2 + -2x2 Combine like terms: 2x + -2x = 0 5 + -5x + -2x2 = 0 + 2x2 + -2x2 5 + -5x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 5 + -5x + -2x2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 2.5x + x2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 2.5x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 2.5x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 2.5x + x2 = 0 + 2.5 2.5x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 2.5x + x2 = 2.5 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 = 2.5 + 1.5625 Reorder the terms: 1.5625 + 2.5x + x2 = 2.5 + 1.5625 Combine like terms: 2.5 + 1.5625 = 4.0625 1.5625 + 2.5x + x2 = 4.0625 Factor a perfect square on the left side: (x + 1.25)(x + 1.25) = 4.0625 Calculate the square root of the right side: 2.015564437 Break this problem into two subproblems by setting (x + 1.25) equal to 2.015564437 and -2.015564437.Subproblem 1
x + 1.25 = 2.015564437 Simplifying x + 1.25 = 2.015564437 Reorder the terms: 1.25 + x = 2.015564437 Solving 1.25 + x = 2.015564437 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 = 2.015564437 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = 2.015564437 + -1.25 x = 2.015564437 + -1.25 Combine like terms: 2.015564437 + -1.25 = 0.765564437 x = 0.765564437 Simplifying x = 0.765564437Subproblem 2
x + 1.25 = -2.015564437 Simplifying x + 1.25 = -2.015564437 Reorder the terms: 1.25 + x = -2.015564437 Solving 1.25 + x = -2.015564437 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 = -2.015564437 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + x = -2.015564437 + -1.25 x = -2.015564437 + -1.25 Combine like terms: -2.015564437 + -1.25 = -3.265564437 x = -3.265564437 Simplifying x = -3.265564437Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.765564437, -3.265564437}
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