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Simplifying 0.1x = -0.2x2 + 10x + 5 Reorder the terms: 0.1x = 5 + 10x + -0.2x2 Solving 0.1x = 5 + 10x + -0.2x2 Solving for variable 'x'. Reorder the terms: -5 + 0.1x + -10x + 0.2x2 = 5 + 10x + -0.2x2 + -5 + -10x + 0.2x2 Combine like terms: 0.1x + -10x = -9.9x -5 + -9.9x + 0.2x2 = 5 + 10x + -0.2x2 + -5 + -10x + 0.2x2 Reorder the terms: -5 + -9.9x + 0.2x2 = 5 + -5 + 10x + -10x + -0.2x2 + 0.2x2 Combine like terms: 5 + -5 = 0 -5 + -9.9x + 0.2x2 = 0 + 10x + -10x + -0.2x2 + 0.2x2 -5 + -9.9x + 0.2x2 = 10x + -10x + -0.2x2 + 0.2x2 Combine like terms: 10x + -10x = 0 -5 + -9.9x + 0.2x2 = 0 + -0.2x2 + 0.2x2 -5 + -9.9x + 0.2x2 = -0.2x2 + 0.2x2 Combine like terms: -0.2x2 + 0.2x2 = 0.0 -5 + -9.9x + 0.2x2 = 0.0 Begin completing the square. Divide all terms by 0.2 the coefficient of the squared term: Divide each side by '0.2'. -25 + -49.5x + x2 = 0 Move the constant term to the right: Add '25' to each side of the equation. -25 + -49.5x + 25 + x2 = 0 + 25 Reorder the terms: -25 + 25 + -49.5x + x2 = 0 + 25 Combine like terms: -25 + 25 = 0 0 + -49.5x + x2 = 0 + 25 -49.5x + x2 = 0 + 25 Combine like terms: 0 + 25 = 25 -49.5x + x2 = 25 The x term is -49.5x. Take half its coefficient (-24.75). Square it (612.5625) and add it to both sides. Add '612.5625' to each side of the equation. -49.5x + 612.5625 + x2 = 25 + 612.5625 Reorder the terms: 612.5625 + -49.5x + x2 = 25 + 612.5625 Combine like terms: 25 + 612.5625 = 637.5625 612.5625 + -49.5x + x2 = 637.5625 Factor a perfect square on the left side: (x + -24.75)(x + -24.75) = 637.5625 Calculate the square root of the right side: 25.25 Break this problem into two subproblems by setting (x + -24.75) equal to 25.25 and -25.25.Subproblem 1
x + -24.75 = 25.25 Simplifying x + -24.75 = 25.25 Reorder the terms: -24.75 + x = 25.25 Solving -24.75 + x = 25.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24.75' to each side of the equation. -24.75 + 24.75 + x = 25.25 + 24.75 Combine like terms: -24.75 + 24.75 = 0.00 0.00 + x = 25.25 + 24.75 x = 25.25 + 24.75 Combine like terms: 25.25 + 24.75 = 50 x = 50 Simplifying x = 50Subproblem 2
x + -24.75 = -25.25 Simplifying x + -24.75 = -25.25 Reorder the terms: -24.75 + x = -25.25 Solving -24.75 + x = -25.25 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '24.75' to each side of the equation. -24.75 + 24.75 + x = -25.25 + 24.75 Combine like terms: -24.75 + 24.75 = 0.00 0.00 + x = -25.25 + 24.75 x = -25.25 + 24.75 Combine like terms: -25.25 + 24.75 = -0.5 x = -0.5 Simplifying x = -0.5Solution
The solution to the problem is based on the solutions from the subproblems. x = {50, -0.5}
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