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Simplifying 1590 = -0.1x2 + 100x + -15000 Reorder the terms: 1590 = -15000 + 100x + -0.1x2 Solving 1590 = -15000 + 100x + -0.1x2 Solving for variable 'x'. Combine like terms: 1590 + 15000 = 16590 16590 + -100x + 0.1x2 = -15000 + 100x + -0.1x2 + 15000 + -100x + 0.1x2 Reorder the terms: 16590 + -100x + 0.1x2 = -15000 + 15000 + 100x + -100x + -0.1x2 + 0.1x2 Combine like terms: -15000 + 15000 = 0 16590 + -100x + 0.1x2 = 0 + 100x + -100x + -0.1x2 + 0.1x2 16590 + -100x + 0.1x2 = 100x + -100x + -0.1x2 + 0.1x2 Combine like terms: 100x + -100x = 0 16590 + -100x + 0.1x2 = 0 + -0.1x2 + 0.1x2 16590 + -100x + 0.1x2 = -0.1x2 + 0.1x2 Combine like terms: -0.1x2 + 0.1x2 = 0.0 16590 + -100x + 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'. 165900 + -1000x + x2 = 0 Move the constant term to the right: Add '-165900' to each side of the equation. 165900 + -1000x + -165900 + x2 = 0 + -165900 Reorder the terms: 165900 + -165900 + -1000x + x2 = 0 + -165900 Combine like terms: 165900 + -165900 = 0 0 + -1000x + x2 = 0 + -165900 -1000x + x2 = 0 + -165900 Combine like terms: 0 + -165900 = -165900 -1000x + x2 = -165900 The x term is -1000x. Take half its coefficient (-500). Square it (250000) and add it to both sides. Add '250000' to each side of the equation. -1000x + 250000 + x2 = -165900 + 250000 Reorder the terms: 250000 + -1000x + x2 = -165900 + 250000 Combine like terms: -165900 + 250000 = 84100 250000 + -1000x + x2 = 84100 Factor a perfect square on the left side: (x + -500)(x + -500) = 84100 Calculate the square root of the right side: 290 Break this problem into two subproblems by setting (x + -500) equal to 290 and -290.Subproblem 1
x + -500 = 290 Simplifying x + -500 = 290 Reorder the terms: -500 + x = 290 Solving -500 + x = 290 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + x = 290 + 500 Combine like terms: -500 + 500 = 0 0 + x = 290 + 500 x = 290 + 500 Combine like terms: 290 + 500 = 790 x = 790 Simplifying x = 790Subproblem 2
x + -500 = -290 Simplifying x + -500 = -290 Reorder the terms: -500 + x = -290 Solving -500 + x = -290 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + x = -290 + 500 Combine like terms: -500 + 500 = 0 0 + x = -290 + 500 x = -290 + 500 Combine like terms: -290 + 500 = 210 x = 210 Simplifying x = 210Solution
The solution to the problem is based on the solutions from the subproblems. x = {790, 210}
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