45*(r)(r-1)+5=0

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Solution for 45*(r)(r-1)+5=0 equation: 


Simplifying
45(r)(r + -1) + 5 = 0

Reorder the terms:
45r(-1 + r) + 5 = 0
(-1 * 45r + r * 45r) + 5 = 0
(-45r + 45r2) + 5 = 0

Reorder the terms:
5 + -45r + 45r2 = 0

Solving
5 + -45r + 45r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '5'.
5(1 + -9r + 9r2) = 0

Ignore the factor 5.

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

Simplifying
1 + -9r + 9r2 = 0

Solving
1 + -9r + 9r2 = 0

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

Divide each side by '9'.
0.1111111111 + -1r + r2 = 0

Move the constant term to the right:

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -1r + -0.1111111111 + r2 = 0 + -0.1111111111

Reorder the terms:
0.1111111111 + -0.1111111111 + -1r + r2 = 0 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + -1r + r2 = 0 + -0.1111111111
-1r + r2 = 0 + -0.1111111111

Combine like terms: 0 + -0.1111111111 = -0.1111111111
-1r + r2 = -0.1111111111

The r term is -1r.  Take half its coefficient (-0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
-1r + 0.25 + r2 = -0.1111111111 + 0.25

Reorder the terms:
0.25 + -1r + r2 = -0.1111111111 + 0.25

Combine like terms: -0.1111111111 + 0.25 = 0.1388888889
0.25 + -1r + r2 = 0.1388888889

Factor a perfect square on the left side:
(r + -0.5)(r + -0.5) = 0.1388888889

Calculate the square root of the right side: 0.372677996

Break this problem into two subproblems by setting 
(r + -0.5) equal to 0.372677996 and -0.372677996.


Subproblem 1
r + -0.5 = 0.372677996

Simplifying
r + -0.5 = 0.372677996

Reorder the terms:
-0.5 + r = 0.372677996

Solving
-0.5 + r = 0.372677996

Solving for variable 'r'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + r = 0.372677996 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + r = 0.372677996 + 0.5
r = 0.372677996 + 0.5

Combine like terms: 0.372677996 + 0.5 = 0.872677996
r = 0.872677996

Simplifying
r = 0.872677996


Subproblem 2
r + -0.5 = -0.372677996

Simplifying
r + -0.5 = -0.372677996

Reorder the terms:
-0.5 + r = -0.372677996

Solving
-0.5 + r = -0.372677996

Solving for variable 'r'.

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

Add '0.5' to each side of the equation.
-0.5 + 0.5 + r = -0.372677996 + 0.5

Combine like terms: -0.5 + 0.5 = 0.0
0.0 + r = -0.372677996 + 0.5
r = -0.372677996 + 0.5

Combine like terms: -0.372677996 + 0.5 = 0.127322004
r = 0.127322004

Simplifying
r = 0.127322004


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {0.872677996, 0.127322004}
Solution
r = {0.872677996, 0.127322004}

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