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Simplifying 105x = 0.2x2 + 10x + 2500 Reorder the terms: 105x = 2500 + 10x + 0.2x2 Solving 105x = 2500 + 10x + 0.2x2 Solving for variable 'x'. Reorder the terms: -2500 + 105x + -10x + -0.2x2 = 2500 + 10x + 0.2x2 + -2500 + -10x + -0.2x2 Combine like terms: 105x + -10x = 95x -2500 + 95x + -0.2x2 = 2500 + 10x + 0.2x2 + -2500 + -10x + -0.2x2 Reorder the terms: -2500 + 95x + -0.2x2 = 2500 + -2500 + 10x + -10x + 0.2x2 + -0.2x2 Combine like terms: 2500 + -2500 = 0 -2500 + 95x + -0.2x2 = 0 + 10x + -10x + 0.2x2 + -0.2x2 -2500 + 95x + -0.2x2 = 10x + -10x + 0.2x2 + -0.2x2 Combine like terms: 10x + -10x = 0 -2500 + 95x + -0.2x2 = 0 + 0.2x2 + -0.2x2 -2500 + 95x + -0.2x2 = 0.2x2 + -0.2x2 Combine like terms: 0.2x2 + -0.2x2 = 0.0 -2500 + 95x + -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'. 12500 + -475x + x2 = 0 Move the constant term to the right: Add '-12500' to each side of the equation. 12500 + -475x + -12500 + x2 = 0 + -12500 Reorder the terms: 12500 + -12500 + -475x + x2 = 0 + -12500 Combine like terms: 12500 + -12500 = 0 0 + -475x + x2 = 0 + -12500 -475x + x2 = 0 + -12500 Combine like terms: 0 + -12500 = -12500 -475x + x2 = -12500 The x term is -475x. Take half its coefficient (-237.5). Square it (56406.25) and add it to both sides. Add '56406.25' to each side of the equation. -475x + 56406.25 + x2 = -12500 + 56406.25 Reorder the terms: 56406.25 + -475x + x2 = -12500 + 56406.25 Combine like terms: -12500 + 56406.25 = 43906.25 56406.25 + -475x + x2 = 43906.25 Factor a perfect square on the left side: (x + -237.5)(x + -237.5) = 43906.25 Calculate the square root of the right side: 209.538182678 Break this problem into two subproblems by setting (x + -237.5) equal to 209.538182678 and -209.538182678.Subproblem 1
x + -237.5 = 209.538182678 Simplifying x + -237.5 = 209.538182678 Reorder the terms: -237.5 + x = 209.538182678 Solving -237.5 + x = 209.538182678 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '237.5' to each side of the equation. -237.5 + 237.5 + x = 209.538182678 + 237.5 Combine like terms: -237.5 + 237.5 = 0.0 0.0 + x = 209.538182678 + 237.5 x = 209.538182678 + 237.5 Combine like terms: 209.538182678 + 237.5 = 447.038182678 x = 447.038182678 Simplifying x = 447.038182678Subproblem 2
x + -237.5 = -209.538182678 Simplifying x + -237.5 = -209.538182678 Reorder the terms: -237.5 + x = -209.538182678 Solving -237.5 + x = -209.538182678 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '237.5' to each side of the equation. -237.5 + 237.5 + x = -209.538182678 + 237.5 Combine like terms: -237.5 + 237.5 = 0.0 0.0 + x = -209.538182678 + 237.5 x = -209.538182678 + 237.5 Combine like terms: -209.538182678 + 237.5 = 27.961817322 x = 27.961817322 Simplifying x = 27.961817322Solution
The solution to the problem is based on the solutions from the subproblems. x = {447.038182678, 27.961817322}
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