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Simplifying x + 0.5 = x * 0.18x Reorder the terms: 0.5 + x = x * 0.18x Reorder the terms for easier multiplication: 0.5 + x = 0.18x * x Multiply x * x 0.5 + x = 0.18x2 Solving 0.5 + x = 0.18x2 Solving for variable 'x'. Combine like terms: 0.18x2 + -0.18x2 = 0.00 0.5 + x + -0.18x2 = 0.00 Begin completing the square. Divide all terms by -0.18 the coefficient of the squared term: Divide each side by '-0.18'. -2.777777778 + -5.555555556x + x2 = 0 Move the constant term to the right: Add '2.777777778' to each side of the equation. -2.777777778 + -5.555555556x + 2.777777778 + x2 = 0 + 2.777777778 Reorder the terms: -2.777777778 + 2.777777778 + -5.555555556x + x2 = 0 + 2.777777778 Combine like terms: -2.777777778 + 2.777777778 = 0.000000000 0.000000000 + -5.555555556x + x2 = 0 + 2.777777778 -5.555555556x + x2 = 0 + 2.777777778 Combine like terms: 0 + 2.777777778 = 2.777777778 -5.555555556x + x2 = 2.777777778 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. -5.555555556x + 0.25 + x2 = 2.777777778 + 0.25 Reorder the terms: 0.25 + -5.555555556x + x2 = 2.777777778 + 0.25 Combine like terms: 2.777777778 + 0.25 = 3.027777778 0.25 + -5.555555556x + x2 = 3.027777778 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 3.027777778 Calculate the square root of the right side: 1.740051085 Break this problem into two subproblems by setting (x + 0.5) equal to 1.740051085 and -1.740051085.Subproblem 1
x + 0.5 = 1.740051085 Simplifying x + 0.5 = 1.740051085 Reorder the terms: 0.5 + x = 1.740051085 Solving 0.5 + x = 1.740051085 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 = 1.740051085 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 1.740051085 + -0.5 x = 1.740051085 + -0.5 Combine like terms: 1.740051085 + -0.5 = 1.240051085 x = 1.240051085 Simplifying x = 1.240051085Subproblem 2
x + 0.5 = -1.740051085 Simplifying x + 0.5 = -1.740051085 Reorder the terms: 0.5 + x = -1.740051085 Solving 0.5 + x = -1.740051085 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 = -1.740051085 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -1.740051085 + -0.5 x = -1.740051085 + -0.5 Combine like terms: -1.740051085 + -0.5 = -2.240051085 x = -2.240051085 Simplifying x = -2.240051085Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.240051085, -2.240051085}
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