(r-6)(r-7)=5

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Solution for (r-6)(r-7)=5 equation: 


Simplifying
(r + -6)(r + -7) = 5

Reorder the terms:
(-6 + r)(r + -7) = 5

Reorder the terms:
(-6 + r)(-7 + r) = 5

Multiply (-6 + r) * (-7 + r)
(-6(-7 + r) + r(-7 + r)) = 5
((-7 * -6 + r * -6) + r(-7 + r)) = 5
((42 + -6r) + r(-7 + r)) = 5
(42 + -6r + (-7 * r + r * r)) = 5
(42 + -6r + (-7r + r2)) = 5

Combine like terms: -6r + -7r = -13r
(42 + -13r + r2) = 5

Solving
42 + -13r + r2 = 5

Solving for variable 'r'.

Reorder the terms:
42 + -5 + -13r + r2 = 5 + -5

Combine like terms: 42 + -5 = 37
37 + -13r + r2 = 5 + -5

Combine like terms: 5 + -5 = 0
37 + -13r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '-37' to each side of the equation.
37 + -13r + -37 + r2 = 0 + -37

Reorder the terms:
37 + -37 + -13r + r2 = 0 + -37

Combine like terms: 37 + -37 = 0
0 + -13r + r2 = 0 + -37
-13r + r2 = 0 + -37

Combine like terms: 0 + -37 = -37
-13r + r2 = -37

The r term is -13r.  Take half its coefficient (-6.5).
Square it (42.25) and add it to both sides.

Add '42.25' to each side of the equation.
-13r + 42.25 + r2 = -37 + 42.25

Reorder the terms:
42.25 + -13r + r2 = -37 + 42.25

Combine like terms: -37 + 42.25 = 5.25
42.25 + -13r + r2 = 5.25

Factor a perfect square on the left side:
(r + -6.5)(r + -6.5) = 5.25

Calculate the square root of the right side: 2.291287847

Break this problem into two subproblems by setting 
(r + -6.5) equal to 2.291287847 and -2.291287847.


Subproblem 1
r + -6.5 = 2.291287847

Simplifying
r + -6.5 = 2.291287847

Reorder the terms:
-6.5 + r = 2.291287847

Solving
-6.5 + r = 2.291287847

Solving for variable 'r'.

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

Add '6.5' to each side of the equation.
-6.5 + 6.5 + r = 2.291287847 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + r = 2.291287847 + 6.5
r = 2.291287847 + 6.5

Combine like terms: 2.291287847 + 6.5 = 8.791287847
r = 8.791287847

Simplifying
r = 8.791287847


Subproblem 2
r + -6.5 = -2.291287847

Simplifying
r + -6.5 = -2.291287847

Reorder the terms:
-6.5 + r = -2.291287847

Solving
-6.5 + r = -2.291287847

Solving for variable 'r'.

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

Add '6.5' to each side of the equation.
-6.5 + 6.5 + r = -2.291287847 + 6.5

Combine like terms: -6.5 + 6.5 = 0.0
0.0 + r = -2.291287847 + 6.5
r = -2.291287847 + 6.5

Combine like terms: -2.291287847 + 6.5 = 4.208712153
r = 4.208712153

Simplifying
r = 4.208712153


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {8.791287847, 4.208712153}

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