8038.4=401.92r+6.28r^2

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Solution for 8038.4=401.92r+6.28r^2 equation:


Simplifying
8038.4 = 401.92r + 6.28r2

Solving
8038.4 = 401.92r + 6.28r2

Solving for variable 'r'.

Reorder the terms:
8038.4 + -401.92r + -6.28r2 = 401.92r + -401.92r + 6.28r2 + -6.28r2

Combine like terms: 401.92r + -401.92r = 0.00
8038.4 + -401.92r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2
8038.4 + -401.92r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
8038.4 + -401.92r + -6.28r2 = 0.00

Begin completing the square.  Divide all terms by
-6.28 the coefficient of the squared term: 

Divide each side by '-6.28'.
-1280 + 64r + r2 = 0

Move the constant term to the right:

Add '1280' to each side of the equation.
-1280 + 64r + 1280 + r2 = 0 + 1280

Reorder the terms:
-1280 + 1280 + 64r + r2 = 0 + 1280

Combine like terms: -1280 + 1280 = 0
0 + 64r + r2 = 0 + 1280
64r + r2 = 0 + 1280

Combine like terms: 0 + 1280 = 1280
64r + r2 = 1280

The r term is 64r.  Take half its coefficient (32).
Square it (1024) and add it to both sides.

Add '1024' to each side of the equation.
64r + 1024 + r2 = 1280 + 1024

Reorder the terms:
1024 + 64r + r2 = 1280 + 1024

Combine like terms: 1280 + 1024 = 2304
1024 + 64r + r2 = 2304

Factor a perfect square on the left side:
(r + 32)(r + 32) = 2304

Calculate the square root of the right side: 48

Break this problem into two subproblems by setting 
(r + 32) equal to 48 and -48.

Subproblem 1

r + 32 = 48 Simplifying r + 32 = 48 Reorder the terms: 32 + r = 48 Solving 32 + r = 48 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-32' to each side of the equation. 32 + -32 + r = 48 + -32 Combine like terms: 32 + -32 = 0 0 + r = 48 + -32 r = 48 + -32 Combine like terms: 48 + -32 = 16 r = 16 Simplifying r = 16

Subproblem 2

r + 32 = -48 Simplifying r + 32 = -48 Reorder the terms: 32 + r = -48 Solving 32 + r = -48 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-32' to each side of the equation. 32 + -32 + r = -48 + -32 Combine like terms: 32 + -32 = 0 0 + r = -48 + -32 r = -48 + -32 Combine like terms: -48 + -32 = -80 r = -80 Simplifying r = -80

Solution

The solution to the problem is based on the solutions from the subproblems. r = {16, -80}

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