0=6.28r^2+2028.44r-47533.32

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Solution for 0=6.28r^2+2028.44r-47533.32 equation:


Simplifying
0 = 6.28r2 + 2028.44r + -47533.32

Reorder the terms:
0 = -47533.32 + 2028.44r + 6.28r2

Solving
0 = -47533.32 + 2028.44r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 47533.32 = 47533.32
47533.32 + -2028.44r + -6.28r2 = -47533.32 + 2028.44r + 6.28r2 + 47533.32 + -2028.44r + -6.28r2

Reorder the terms:
47533.32 + -2028.44r + -6.28r2 = -47533.32 + 47533.32 + 2028.44r + -2028.44r + 6.28r2 + -6.28r2

Combine like terms: -47533.32 + 47533.32 = 0.00
47533.32 + -2028.44r + -6.28r2 = 0.00 + 2028.44r + -2028.44r + 6.28r2 + -6.28r2
47533.32 + -2028.44r + -6.28r2 = 2028.44r + -2028.44r + 6.28r2 + -6.28r2

Combine like terms: 2028.44r + -2028.44r = 0.00
47533.32 + -2028.44r + -6.28r2 = 0.00 + 6.28r2 + -6.28r2
47533.32 + -2028.44r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
47533.32 + -2028.44r + -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'.
-7569 + 323r + r2 = 0

Move the constant term to the right:

Add '7569' to each side of the equation.
-7569 + 323r + 7569 + r2 = 0 + 7569

Reorder the terms:
-7569 + 7569 + 323r + r2 = 0 + 7569

Combine like terms: -7569 + 7569 = 0
0 + 323r + r2 = 0 + 7569
323r + r2 = 0 + 7569

Combine like terms: 0 + 7569 = 7569
323r + r2 = 7569

The r term is 323r.  Take half its coefficient (161.5).
Square it (26082.25) and add it to both sides.

Add '26082.25' to each side of the equation.
323r + 26082.25 + r2 = 7569 + 26082.25

Reorder the terms:
26082.25 + 323r + r2 = 7569 + 26082.25

Combine like terms: 7569 + 26082.25 = 33651.25
26082.25 + 323r + r2 = 33651.25

Factor a perfect square on the left side:
(r + 161.5)(r + 161.5) = 33651.25

Calculate the square root of the right side: 183.442770367

Break this problem into two subproblems by setting 
(r + 161.5) equal to 183.442770367 and -183.442770367.

Subproblem 1

r + 161.5 = 183.442770367 Simplifying r + 161.5 = 183.442770367 Reorder the terms: 161.5 + r = 183.442770367 Solving 161.5 + r = 183.442770367 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-161.5' to each side of the equation. 161.5 + -161.5 + r = 183.442770367 + -161.5 Combine like terms: 161.5 + -161.5 = 0.0 0.0 + r = 183.442770367 + -161.5 r = 183.442770367 + -161.5 Combine like terms: 183.442770367 + -161.5 = 21.942770367 r = 21.942770367 Simplifying r = 21.942770367

Subproblem 2

r + 161.5 = -183.442770367 Simplifying r + 161.5 = -183.442770367 Reorder the terms: 161.5 + r = -183.442770367 Solving 161.5 + r = -183.442770367 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-161.5' to each side of the equation. 161.5 + -161.5 + r = -183.442770367 + -161.5 Combine like terms: 161.5 + -161.5 = 0.0 0.0 + r = -183.442770367 + -161.5 r = -183.442770367 + -161.5 Combine like terms: -183.442770367 + -161.5 = -344.942770367 r = -344.942770367 Simplifying r = -344.942770367

Solution

The solution to the problem is based on the solutions from the subproblems. r = {21.942770367, -344.942770367}

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