a^2-1000a+52546.5=0

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Solution for a^2-1000a+52546.5=0 equation:


Simplifying
a2 + -1000a + 52546.5 = 0

Reorder the terms:
52546.5 + -1000a + a2 = 0

Solving
52546.5 + -1000a + a2 = 0

Solving for variable 'a'.

Begin completing the square.

Move the constant term to the right:

Add '-52546.5' to each side of the equation.
52546.5 + -1000a + -52546.5 + a2 = 0 + -52546.5

Reorder the terms:
52546.5 + -52546.5 + -1000a + a2 = 0 + -52546.5

Combine like terms: 52546.5 + -52546.5 = 0.0
0.0 + -1000a + a2 = 0 + -52546.5
-1000a + a2 = 0 + -52546.5

Combine like terms: 0 + -52546.5 = -52546.5
-1000a + a2 = -52546.5

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 = -52546.5 + 250000

Reorder the terms:
250000 + -1000a + a2 = -52546.5 + 250000

Combine like terms: -52546.5 + 250000 = 197453.5
250000 + -1000a + a2 = 197453.5

Factor a perfect square on the left side:
(a + -500)(a + -500) = 197453.5

Calculate the square root of the right side: 444.357401199

Break this problem into two subproblems by setting 
(a + -500) equal to 444.357401199 and -444.357401199.


Subproblem 1
a + -500 = 444.357401199

Simplifying
a + -500 = 444.357401199

Reorder the terms:
-500 + a = 444.357401199

Solving
-500 + a = 444.357401199

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 = 444.357401199 + 500

Combine like terms: -500 + 500 = 0
0 + a = 444.357401199 + 500
a = 444.357401199 + 500

Combine like terms: 444.357401199 + 500 = 944.357401199
a = 944.357401199

Simplifying
a = 944.357401199


Subproblem 2
a + -500 = -444.357401199

Simplifying
a + -500 = -444.357401199

Reorder the terms:
-500 + a = -444.357401199

Solving
-500 + a = -444.357401199

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 = -444.357401199 + 500

Combine like terms: -500 + 500 = 0
0 + a = -444.357401199 + 500
a = -444.357401199 + 500

Combine like terms: -444.357401199 + 500 = 55.642598801
a = 55.642598801

Simplifying
a = 55.642598801


Solution
The solution to the problem is based on the solutions
from the subproblems.
a = {944.357401199, 55.642598801}

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