(r-3)(r+7)=-13

Solution for (r-3)(r+7)=-13 equation:

Simplifying
(r + -3)(r + 7) = -13

Reorder the terms:
(-3 + r)(r + 7) = -13

Reorder the terms:
(-3 + r)(7 + r) = -13

Multiply (-3 + r) * (7 + r)
(-3(7 + r) + r(7 + r)) = -13
((7 * -3 + r * -3) + r(7 + r)) = -13
((-21 + -3r) + r(7 + r)) = -13
(-21 + -3r + (7 * r + r * r)) = -13
(-21 + -3r + (7r + r2)) = -13

Combine like terms: -3r + 7r = 4r
(-21 + 4r + r2) = -13

Solving
-21 + 4r + r2 = -13

Solving for variable 'r'.

Reorder the terms:
-21 + 13 + 4r + r2 = -13 + 13

Combine like terms: -21 + 13 = -8
-8 + 4r + r2 = -13 + 13

Combine like terms: -13 + 13 = 0
-8 + 4r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '8' to each side of the equation.
-8 + 4r + 8 + r2 = 0 + 8

Reorder the terms:
-8 + 8 + 4r + r2 = 0 + 8

Combine like terms: -8 + 8 = 0
0 + 4r + r2 = 0 + 8
4r + r2 = 0 + 8

Combine like terms: 0 + 8 = 8
4r + r2 = 8

The r term is 4r.  Take half its coefficient (2).
Square it (4) and add it to both sides.

Add '4' to each side of the equation.
4r + 4 + r2 = 8 + 4

Reorder the terms:
4 + 4r + r2 = 8 + 4

Combine like terms: 8 + 4 = 12
4 + 4r + r2 = 12

Factor a perfect square on the left side:
(r + 2)(r + 2) = 12

Calculate the square root of the right side: 3.464101615

Break this problem into two subproblems by setting 
(r + 2) equal to 3.464101615 and -3.464101615.

Subproblem 1
r + 2 = 3.464101615

Simplifying
r + 2 = 3.464101615

Reorder the terms:
2 + r = 3.464101615

Solving
2 + r = 3.464101615

Solving for variable 'r'.

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

Add '-2' to each side of the equation.
2 + -2 + r = 3.464101615 + -2

Combine like terms: 2 + -2 = 0
0 + r = 3.464101615 + -2
r = 3.464101615 + -2

Combine like terms: 3.464101615 + -2 = 1.464101615
r = 1.464101615

Simplifying
r = 1.464101615

Subproblem 2
r + 2 = -3.464101615

Simplifying
r + 2 = -3.464101615

Reorder the terms:
2 + r = -3.464101615

Solving
2 + r = -3.464101615

Solving for variable 'r'.

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

Add '-2' to each side of the equation.
2 + -2 + r = -3.464101615 + -2

Combine like terms: 2 + -2 = 0
0 + r = -3.464101615 + -2
r = -3.464101615 + -2

Combine like terms: -3.464101615 + -2 = -5.464101615
r = -5.464101615

Simplifying
r = -5.464101615

Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.464101615, -5.464101615}