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Simplifying 0.5x2 + 0.5x = 45 Reorder the terms: 0.5x + 0.5x2 = 45 Solving 0.5x + 0.5x2 = 45 Solving for variable 'x'. Reorder the terms: -45 + 0.5x + 0.5x2 = 45 + -45 Combine like terms: 45 + -45 = 0 -45 + 0.5x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -90 + x + x2 = 0 Move the constant term to the right: Add '90' to each side of the equation. -90 + x + 90 + x2 = 0 + 90 Reorder the terms: -90 + 90 + x + x2 = 0 + 90 Combine like terms: -90 + 90 = 0 0 + x + x2 = 0 + 90 x + x2 = 0 + 90 Combine like terms: 0 + 90 = 90 x + x2 = 90 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. + 0.25 + x2 = 90 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 90 + 0.25 Combine like terms: 90 + 0.25 = 90.25 1.25 + x2 = 90.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 90.25 Calculate the square root of the right side: 9.5 Break this problem into two subproblems by setting (x + 0.5) equal to 9.5 and -9.5.Subproblem 1
x + 0.5 = 9.5 Simplifying x + 0.5 = 9.5 Reorder the terms: 0.5 + x = 9.5 Solving 0.5 + x = 9.5 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 = 9.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 9.5 + -0.5 x = 9.5 + -0.5 Combine like terms: 9.5 + -0.5 = 9 x = 9 Simplifying x = 9Subproblem 2
x + 0.5 = -9.5 Simplifying x + 0.5 = -9.5 Reorder the terms: 0.5 + x = -9.5 Solving 0.5 + x = -9.5 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 = -9.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -9.5 + -0.5 x = -9.5 + -0.5 Combine like terms: -9.5 + -0.5 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {9, -10}
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