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Simplifying 0.5x2 + x = 10 Reorder the terms: x + 0.5x2 = 10 Solving x + 0.5x2 = 10 Solving for variable 'x'. Reorder the terms: -10 + x + 0.5x2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + x + 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'. -20 + 2x + x2 = 0 Move the constant term to the right: Add '20' to each side of the equation. -20 + 2x + 20 + x2 = 0 + 20 Reorder the terms: -20 + 20 + 2x + x2 = 0 + 20 Combine like terms: -20 + 20 = 0 0 + 2x + x2 = 0 + 20 2x + x2 = 0 + 20 Combine like terms: 0 + 20 = 20 2x + x2 = 20 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. 2x + 0.25 + x2 = 20 + 0.25 Reorder the terms: 0.25 + 2x + x2 = 20 + 0.25 Combine like terms: 20 + 0.25 = 20.25 0.25 + 2x + x2 = 20.25 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 20.25 Calculate the square root of the right side: 4.5 Break this problem into two subproblems by setting (x + 0.5) equal to 4.5 and -4.5.Subproblem 1
x + 0.5 = 4.5 Simplifying x + 0.5 = 4.5 Reorder the terms: 0.5 + x = 4.5 Solving 0.5 + x = 4.5 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 = 4.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 4.5 + -0.5 x = 4.5 + -0.5 Combine like terms: 4.5 + -0.5 = 4 x = 4 Simplifying x = 4Subproblem 2
x + 0.5 = -4.5 Simplifying x + 0.5 = -4.5 Reorder the terms: 0.5 + x = -4.5 Solving 0.5 + x = -4.5 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 = -4.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -4.5 + -0.5 x = -4.5 + -0.5 Combine like terms: -4.5 + -0.5 = -5 x = -5 Simplifying x = -5Solution
The solution to the problem is based on the solutions from the subproblems. x = {4, -5}
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