a^2-1000a+40182.6=0

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Solution for a^2-1000a+40182.6=0 equation:


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 = 958.058293234

Subproblem 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 = 41.941706766

Solution

The solution to the problem is based on the solutions from the subproblems. a = {958.058293234, 41.941706766}

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