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Simplifying a2 + -1000a + 40182.6 = 0 Reorder the terms: 40182.6 + -1000a + a2 = 0 Solving 40182.6 + -1000a + a2 = 0 Solving for variable 'a'. Begin completing the square. Move the constant term to the right: Add '-40182.6' to each side of the equation. 40182.6 + -1000a + -40182.6 + a2 = 0 + -40182.6 Reorder the terms: 40182.6 + -40182.6 + -1000a + a2 = 0 + -40182.6 Combine like terms: 40182.6 + -40182.6 = 0.0 0.0 + -1000a + a2 = 0 + -40182.6 -1000a + a2 = 0 + -40182.6 Combine like terms: 0 + -40182.6 = -40182.6 -1000a + a2 = -40182.6 The a term is -1000a. Take half its coefficient (-500). Square it (250000) and add it to both sides. Add '250000' to each side of the equation. -1000a + 250000 + a2 = -40182.6 + 250000 Reorder the terms: 250000 + -1000a + a2 = -40182.6 + 250000 Combine like terms: -40182.6 + 250000 = 209817.4 250000 + -1000a + a2 = 209817.4 Factor a perfect square on the left side: (a + -500)(a + -500) = 209817.4 Calculate the square root of the right side: 458.058293234 Break this problem into two subproblems by setting (a + -500) equal to 458.058293234 and -458.058293234.Subproblem 1
a + -500 = 458.058293234 Simplifying a + -500 = 458.058293234 Reorder the terms: -500 + a = 458.058293234 Solving -500 + a = 458.058293234 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + a = 458.058293234 + 500 Combine like terms: -500 + 500 = 0 0 + a = 458.058293234 + 500 a = 458.058293234 + 500 Combine like terms: 458.058293234 + 500 = 958.058293234 a = 958.058293234 Simplifying a = 958.058293234Subproblem 2
a + -500 = -458.058293234 Simplifying a + -500 = -458.058293234 Reorder the terms: -500 + a = -458.058293234 Solving -500 + a = -458.058293234 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '500' to each side of the equation. -500 + 500 + a = -458.058293234 + 500 Combine like terms: -500 + 500 = 0 0 + a = -458.058293234 + 500 a = -458.058293234 + 500 Combine like terms: -458.058293234 + 500 = 41.941706766 a = 41.941706766 Simplifying a = 41.941706766Solution
The solution to the problem is based on the solutions from the subproblems. a = {958.058293234, 41.941706766}
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