10r^2-450r+45=0

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Solution for 10r^2-450r+45=0 equation: 


Simplifying
10r2 + -450r + 45 = 0

Reorder the terms:
45 + -450r + 10r2 = 0

Solving
45 + -450r + 10r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '5'.
5(9 + -90r + 2r2) = 0

Ignore the factor 5.

Subproblem 1
Set the factor '(9 + -90r + 2r2)' equal to zero and attempt to solve:

Simplifying
9 + -90r + 2r2 = 0

Solving
9 + -90r + 2r2 = 0

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

Divide each side by '2'.
4.5 + -45r + r2 = 0

Move the constant term to the right:

Add '-4.5' to each side of the equation.
4.5 + -45r + -4.5 + r2 = 0 + -4.5

Reorder the terms:
4.5 + -4.5 + -45r + r2 = 0 + -4.5

Combine like terms: 4.5 + -4.5 = 0.0
0.0 + -45r + r2 = 0 + -4.5
-45r + r2 = 0 + -4.5

Combine like terms: 0 + -4.5 = -4.5
-45r + r2 = -4.5

The r term is -45r.  Take half its coefficient (-22.5).
Square it (506.25) and add it to both sides.

Add '506.25' to each side of the equation.
-45r + 506.25 + r2 = -4.5 + 506.25

Reorder the terms:
506.25 + -45r + r2 = -4.5 + 506.25

Combine like terms: -4.5 + 506.25 = 501.75
506.25 + -45r + r2 = 501.75

Factor a perfect square on the left side:
(r + -22.5)(r + -22.5) = 501.75

Calculate the square root of the right side: 22.399776785

Break this problem into two subproblems by setting 
(r + -22.5) equal to 22.399776785 and -22.399776785.


Subproblem 1
r + -22.5 = 22.399776785

Simplifying
r + -22.5 = 22.399776785

Reorder the terms:
-22.5 + r = 22.399776785

Solving
-22.5 + r = 22.399776785

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '22.5' to each side of the equation.
-22.5 + 22.5 + r = 22.399776785 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + r = 22.399776785 + 22.5
r = 22.399776785 + 22.5

Combine like terms: 22.399776785 + 22.5 = 44.899776785
r = 44.899776785

Simplifying
r = 44.899776785


Subproblem 2
r + -22.5 = -22.399776785

Simplifying
r + -22.5 = -22.399776785

Reorder the terms:
-22.5 + r = -22.399776785

Solving
-22.5 + r = -22.399776785

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '22.5' to each side of the equation.
-22.5 + 22.5 + r = -22.399776785 + 22.5

Combine like terms: -22.5 + 22.5 = 0.0
0.0 + r = -22.399776785 + 22.5
r = -22.399776785 + 22.5

Combine like terms: -22.399776785 + 22.5 = 0.100223215
r = 0.100223215

Simplifying
r = 0.100223215


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {44.899776785, 0.100223215}
Solution
r = {44.899776785, 0.100223215}

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