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Simplifying 20 = 0.1x2 + 0.5x + 15 Reorder the terms: 20 = 15 + 0.5x + 0.1x2 Solving 20 = 15 + 0.5x + 0.1x2 Solving for variable 'x'. Combine like terms: 20 + -15 = 5 5 + -0.5x + -0.1x2 = 15 + 0.5x + 0.1x2 + -15 + -0.5x + -0.1x2 Reorder the terms: 5 + -0.5x + -0.1x2 = 15 + -15 + 0.5x + -0.5x + 0.1x2 + -0.1x2 Combine like terms: 15 + -15 = 0 5 + -0.5x + -0.1x2 = 0 + 0.5x + -0.5x + 0.1x2 + -0.1x2 5 + -0.5x + -0.1x2 = 0.5x + -0.5x + 0.1x2 + -0.1x2 Combine like terms: 0.5x + -0.5x = 0.0 5 + -0.5x + -0.1x2 = 0.0 + 0.1x2 + -0.1x2 5 + -0.5x + -0.1x2 = 0.1x2 + -0.1x2 Combine like terms: 0.1x2 + -0.1x2 = 0.0 5 + -0.5x + -0.1x2 = 0.0 Begin completing the square. Divide all terms by -0.1 the coefficient of the squared term: Divide each side by '-0.1'. -50 + 5x + x2 = 0 Move the constant term to the right: Add '50' to each side of the equation. -50 + 5x + 50 + x2 = 0 + 50 Reorder the terms: -50 + 50 + 5x + x2 = 0 + 50 Combine like terms: -50 + 50 = 0 0 + 5x + x2 = 0 + 50 5x + x2 = 0 + 50 Combine like terms: 0 + 50 = 50 5x + x2 = 50 The x term is 5x. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5x + 6.25 + x2 = 50 + 6.25 Reorder the terms: 6.25 + 5x + x2 = 50 + 6.25 Combine like terms: 50 + 6.25 = 56.25 6.25 + 5x + x2 = 56.25 Factor a perfect square on the left side: (x + 2.5)(x + 2.5) = 56.25 Calculate the square root of the right side: 7.5 Break this problem into two subproblems by setting (x + 2.5) equal to 7.5 and -7.5.Subproblem 1
x + 2.5 = 7.5 Simplifying x + 2.5 = 7.5 Reorder the terms: 2.5 + x = 7.5 Solving 2.5 + x = 7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = 7.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = 7.5 + -2.5 x = 7.5 + -2.5 Combine like terms: 7.5 + -2.5 = 5 x = 5 Simplifying x = 5Subproblem 2
x + 2.5 = -7.5 Simplifying x + 2.5 = -7.5 Reorder the terms: 2.5 + x = -7.5 Solving 2.5 + x = -7.5 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + x = -7.5 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + x = -7.5 + -2.5 x = -7.5 + -2.5 Combine like terms: -7.5 + -2.5 = -10 x = -10 Simplifying x = -10Solution
The solution to the problem is based on the solutions from the subproblems. x = {5, -10}
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