400a-a^2=15846.47

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Solution for 400a-a^2=15846.47 equation:


Simplifying
400a + -1a2 = 15846.47

Solving
400a + -1a2 = 15846.47

Solving for variable 'a'.

Reorder the terms:
-15846.47 + 400a + -1a2 = 15846.47 + -15846.47

Combine like terms: 15846.47 + -15846.47 = 0.00
-15846.47 + 400a + -1a2 = 0.00

Begin completing the square.  Divide all terms by
-1 the coefficient of the squared term: 

Divide each side by '-1'.
15846.47 + -400a + a2 = 0.00

Move the constant term to the right:

Add '-15846.47' to each side of the equation.
15846.47 + -400a + -15846.47 + a2 = 0.00 + -15846.47

Reorder the terms:
15846.47 + -15846.47 + -400a + a2 = 0.00 + -15846.47

Combine like terms: 15846.47 + -15846.47 = 0.00
0.00 + -400a + a2 = 0.00 + -15846.47
-400a + a2 = 0.00 + -15846.47

Combine like terms: 0.00 + -15846.47 = -15846.47
-400a + a2 = -15846.47

The a term is -400a.  Take half its coefficient (-200).
Square it (40000) and add it to both sides.

Add '40000' to each side of the equation.
-400a + 40000 + a2 = -15846.47 + 40000

Reorder the terms:
40000 + -400a + a2 = -15846.47 + 40000

Combine like terms: -15846.47 + 40000 = 24153.53
40000 + -400a + a2 = 24153.53

Factor a perfect square on the left side:
(a + -200)(a + -200) = 24153.53

Calculate the square root of the right side: 155.414059853

Break this problem into two subproblems by setting 
(a + -200) equal to 155.414059853 and -155.414059853.

Subproblem 1

a + -200 = 155.414059853 Simplifying a + -200 = 155.414059853 Reorder the terms: -200 + a = 155.414059853 Solving -200 + a = 155.414059853 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + a = 155.414059853 + 200 Combine like terms: -200 + 200 = 0 0 + a = 155.414059853 + 200 a = 155.414059853 + 200 Combine like terms: 155.414059853 + 200 = 355.414059853 a = 355.414059853 Simplifying a = 355.414059853

Subproblem 2

a + -200 = -155.414059853 Simplifying a + -200 = -155.414059853 Reorder the terms: -200 + a = -155.414059853 Solving -200 + a = -155.414059853 Solving for variable 'a'. Move all terms containing a to the left, all other terms to the right. Add '200' to each side of the equation. -200 + 200 + a = -155.414059853 + 200 Combine like terms: -200 + 200 = 0 0 + a = -155.414059853 + 200 a = -155.414059853 + 200 Combine like terms: -155.414059853 + 200 = 44.585940147 a = 44.585940147 Simplifying a = 44.585940147

Solution

The solution to the problem is based on the solutions from the subproblems. a = {355.414059853, 44.585940147}

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