0=6.28r^2+33.284r-285.9912

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Solution for 0=6.28r^2+33.284r-285.9912 equation:


Simplifying
0 = 6.28r2 + 33.284r + -285.9912

Reorder the terms:
0 = -285.9912 + 33.284r + 6.28r2

Solving
0 = -285.9912 + 33.284r + 6.28r2

Solving for variable 'r'.

Combine like terms: 0 + 285.9912 = 285.9912
285.9912 + -33.284r + -6.28r2 = -285.9912 + 33.284r + 6.28r2 + 285.9912 + -33.284r + -6.28r2

Reorder the terms:
285.9912 + -33.284r + -6.28r2 = -285.9912 + 285.9912 + 33.284r + -33.284r + 6.28r2 + -6.28r2

Combine like terms: -285.9912 + 285.9912 = 0.0000
285.9912 + -33.284r + -6.28r2 = 0.0000 + 33.284r + -33.284r + 6.28r2 + -6.28r2
285.9912 + -33.284r + -6.28r2 = 33.284r + -33.284r + 6.28r2 + -6.28r2

Combine like terms: 33.284r + -33.284r = 0.000
285.9912 + -33.284r + -6.28r2 = 0.000 + 6.28r2 + -6.28r2
285.9912 + -33.284r + -6.28r2 = 6.28r2 + -6.28r2

Combine like terms: 6.28r2 + -6.28r2 = 0.00
285.9912 + -33.284r + -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'.
-45.54 + 5.3r + r2 = 0

Move the constant term to the right:

Add '45.54' to each side of the equation.
-45.54 + 5.3r + 45.54 + r2 = 0 + 45.54

Reorder the terms:
-45.54 + 45.54 + 5.3r + r2 = 0 + 45.54

Combine like terms: -45.54 + 45.54 = 0.00
0.00 + 5.3r + r2 = 0 + 45.54
5.3r + r2 = 0 + 45.54

Combine like terms: 0 + 45.54 = 45.54
5.3r + r2 = 45.54

The r term is 5.3r.  Take half its coefficient (2.65).
Square it (7.0225) and add it to both sides.

Add '7.0225' to each side of the equation.
5.3r + 7.0225 + r2 = 45.54 + 7.0225

Reorder the terms:
7.0225 + 5.3r + r2 = 45.54 + 7.0225

Combine like terms: 45.54 + 7.0225 = 52.5625
7.0225 + 5.3r + r2 = 52.5625

Factor a perfect square on the left side:
(r + 2.65)(r + 2.65) = 52.5625

Calculate the square root of the right side: 7.25

Break this problem into two subproblems by setting 
(r + 2.65) equal to 7.25 and -7.25.

Subproblem 1

r + 2.65 = 7.25 Simplifying r + 2.65 = 7.25 Reorder the terms: 2.65 + r = 7.25 Solving 2.65 + r = 7.25 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = 7.25 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = 7.25 + -2.65 r = 7.25 + -2.65 Combine like terms: 7.25 + -2.65 = 4.6 r = 4.6 Simplifying r = 4.6

Subproblem 2

r + 2.65 = -7.25 Simplifying r + 2.65 = -7.25 Reorder the terms: 2.65 + r = -7.25 Solving 2.65 + r = -7.25 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.65' to each side of the equation. 2.65 + -2.65 + r = -7.25 + -2.65 Combine like terms: 2.65 + -2.65 = 0.00 0.00 + r = -7.25 + -2.65 r = -7.25 + -2.65 Combine like terms: -7.25 + -2.65 = -9.9 r = -9.9 Simplifying r = -9.9

Solution

The solution to the problem is based on the solutions from the subproblems. r = {4.6, -9.9}

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