7(r^2-7r+1)=0

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Solution for 7(r^2-7r+1)=0 equation: 


Simplifying
7(r2 + -7r + 1) = 0

Reorder the terms:
7(1 + -7r + r2) = 0
(1 * 7 + -7r * 7 + r2 * 7) = 0
(7 + -49r + 7r2) = 0

Solving
7 + -49r + 7r2 = 0

Solving for variable 'r'.

Factor out the Greatest Common Factor (GCF), '7'.
7(1 + -7r + r2) = 0

Ignore the factor 7.

Subproblem 1
Set the factor '(1 + -7r + r2)' equal to zero and attempt to solve:

Simplifying
1 + -7r + r2 = 0

Solving
1 + -7r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + -7r + -1 + r2 = 0 + -1

Reorder the terms:
1 + -1 + -7r + r2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + -7r + r2 = 0 + -1
-7r + r2 = 0 + -1

Combine like terms: 0 + -1 = -1
-7r + r2 = -1

The r term is -7r.  Take half its coefficient (-3.5).
Square it (12.25) and add it to both sides.

Add '12.25' to each side of the equation.
-7r + 12.25 + r2 = -1 + 12.25

Reorder the terms:
12.25 + -7r + r2 = -1 + 12.25

Combine like terms: -1 + 12.25 = 11.25
12.25 + -7r + r2 = 11.25

Factor a perfect square on the left side:
(r + -3.5)(r + -3.5) = 11.25

Calculate the square root of the right side: 3.354101966

Break this problem into two subproblems by setting 
(r + -3.5) equal to 3.354101966 and -3.354101966.


Subproblem 1
r + -3.5 = 3.354101966

Simplifying
r + -3.5 = 3.354101966

Reorder the terms:
-3.5 + r = 3.354101966

Solving
-3.5 + r = 3.354101966

Solving for variable 'r'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + r = 3.354101966 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + r = 3.354101966 + 3.5
r = 3.354101966 + 3.5

Combine like terms: 3.354101966 + 3.5 = 6.854101966
r = 6.854101966

Simplifying
r = 6.854101966


Subproblem 2
r + -3.5 = -3.354101966

Simplifying
r + -3.5 = -3.354101966

Reorder the terms:
-3.5 + r = -3.354101966

Solving
-3.5 + r = -3.354101966

Solving for variable 'r'.

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

Add '3.5' to each side of the equation.
-3.5 + 3.5 + r = -3.354101966 + 3.5

Combine like terms: -3.5 + 3.5 = 0.0
0.0 + r = -3.354101966 + 3.5
r = -3.354101966 + 3.5

Combine like terms: -3.354101966 + 3.5 = 0.145898034
r = 0.145898034

Simplifying
r = 0.145898034


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {6.854101966, 0.145898034}
Solution
r = {6.854101966, 0.145898034}

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