r^2-6r+11=5

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Solution for r^2-6r+11=5 equation: 


Simplifying
r2 + -6r + 11 = 5

Reorder the terms:
11 + -6r + r2 = 5

Solving
11 + -6r + r2 = 5

Solving for variable 'r'.

Reorder the terms:
11 + -5 + -6r + r2 = 5 + -5

Combine like terms: 11 + -5 = 6
6 + -6r + r2 = 5 + -5

Combine like terms: 5 + -5 = 0
6 + -6r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-6' to each side of the equation.
6 + -6r + -6 + r2 = 0 + -6

Reorder the terms:
6 + -6 + -6r + r2 = 0 + -6

Combine like terms: 6 + -6 = 0
0 + -6r + r2 = 0 + -6
-6r + r2 = 0 + -6

Combine like terms: 0 + -6 = -6
-6r + r2 = -6

The r term is -6r.  Take half its coefficient (-3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
-6r + 9 + r2 = -6 + 9

Reorder the terms:
9 + -6r + r2 = -6 + 9

Combine like terms: -6 + 9 = 3
9 + -6r + r2 = 3

Factor a perfect square on the left side:
(r + -3)(r + -3) = 3

Calculate the square root of the right side: 1.732050808

Break this problem into two subproblems by setting 
(r + -3) equal to 1.732050808 and -1.732050808.


Subproblem 1
r + -3 = 1.732050808

Simplifying
r + -3 = 1.732050808

Reorder the terms:
-3 + r = 1.732050808

Solving
-3 + r = 1.732050808

Solving for variable 'r'.

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

Add '3' to each side of the equation.
-3 + 3 + r = 1.732050808 + 3

Combine like terms: -3 + 3 = 0
0 + r = 1.732050808 + 3
r = 1.732050808 + 3

Combine like terms: 1.732050808 + 3 = 4.732050808
r = 4.732050808

Simplifying
r = 4.732050808


Subproblem 2
r + -3 = -1.732050808

Simplifying
r + -3 = -1.732050808

Reorder the terms:
-3 + r = -1.732050808

Solving
-3 + r = -1.732050808

Solving for variable 'r'.

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

Add '3' to each side of the equation.
-3 + 3 + r = -1.732050808 + 3

Combine like terms: -3 + 3 = 0
0 + r = -1.732050808 + 3
r = -1.732050808 + 3

Combine like terms: -1.732050808 + 3 = 1.267949192
r = 1.267949192

Simplifying
r = 1.267949192


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {4.732050808, 1.267949192}

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