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Simplifying 90x + -1(2500 + 10x + 0.2x2) = 0 90x + (2500 * -1 + 10x * -1 + 0.2x2 * -1) = 0 90x + (-2500 + -10x + -0.2x2) = 0 Reorder the terms: -2500 + 90x + -10x + -0.2x2 = 0 Combine like terms: 90x + -10x = 80x -2500 + 80x + -0.2x2 = 0 Solving -2500 + 80x + -0.2x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -0.2 the coefficient of the squared term: Divide each side by '-0.2'. 12500 + -400x + x2 = 0 Move the constant term to the right: Add '-12500' to each side of the equation. 12500 + -400x + -12500 + x2 = 0 + -12500 Reorder the terms: 12500 + -12500 + -400x + x2 = 0 + -12500 Combine like terms: 12500 + -12500 = 0 0 + -400x + x2 = 0 + -12500 -400x + x2 = 0 + -12500 Combine like terms: 0 + -12500 = -12500 -400x + x2 = -12500 The x term is -400x. Take half its coefficient (-200). Square it (40000) and add it to both sides. Add '40000' to each side of the equation. -400x + 40000 + x2 = -12500 + 40000 Reorder the terms: 40000 + -400x + x2 = -12500 + 40000 Combine like terms: -12500 + 40000 = 27500 40000 + -400x + x2 = 27500 Factor a perfect square on the left side: (x + -200)(x + -200) = 27500 Calculate the square root of the right side: 165.831239518 Break this problem into two subproblems by setting (x + -200) equal to 165.831239518 and -165.831239518.Subproblem 1
x + -200 = 165.831239518 Simplifying x + -200 = 165.831239518 Reorder the terms: -200 + x = 165.831239518 Solving -200 + x = 165.831239518 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + x = 165.831239518 + 200 Combine like terms: -200 + 200 = 0 0 + x = 165.831239518 + 200 x = 165.831239518 + 200 Combine like terms: 165.831239518 + 200 = 365.831239518 x = 365.831239518 Simplifying x = 365.831239518Subproblem 2
x + -200 = -165.831239518 Simplifying x + -200 = -165.831239518 Reorder the terms: -200 + x = -165.831239518 Solving -200 + x = -165.831239518 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + x = -165.831239518 + 200 Combine like terms: -200 + 200 = 0 0 + x = -165.831239518 + 200 x = -165.831239518 + 200 Combine like terms: -165.831239518 + 200 = 34.168760482 x = 34.168760482 Simplifying x = 34.168760482Solution
The solution to the problem is based on the solutions from the subproblems. x = {365.831239518, 34.168760482}
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