(r-3)(r+13)=16

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Solution for (r-3)(r+13)=16 equation:


Simplifying
(r + -3)(r + 13) = 16

Reorder the terms:
(-3 + r)(r + 13) = 16

Reorder the terms:
(-3 + r)(13 + r) = 16

Multiply (-3 + r) * (13 + r)
(-3(13 + r) + r(13 + r)) = 16
((13 * -3 + r * -3) + r(13 + r)) = 16
((-39 + -3r) + r(13 + r)) = 16
(-39 + -3r + (13 * r + r * r)) = 16
(-39 + -3r + (13r + r2)) = 16

Combine like terms: -3r + 13r = 10r
(-39 + 10r + r2) = 16

Solving
-39 + 10r + r2 = 16

Solving for variable 'r'.

Reorder the terms:
-39 + -16 + 10r + r2 = 16 + -16

Combine like terms: -39 + -16 = -55
-55 + 10r + r2 = 16 + -16

Combine like terms: 16 + -16 = 0
-55 + 10r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '55' to each side of the equation.
-55 + 10r + 55 + r2 = 0 + 55

Reorder the terms:
-55 + 55 + 10r + r2 = 0 + 55

Combine like terms: -55 + 55 = 0
0 + 10r + r2 = 0 + 55
10r + r2 = 0 + 55

Combine like terms: 0 + 55 = 55
10r + r2 = 55

The r term is 10r.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10r + 25 + r2 = 55 + 25

Reorder the terms:
25 + 10r + r2 = 55 + 25

Combine like terms: 55 + 25 = 80
25 + 10r + r2 = 80

Factor a perfect square on the left side:
(r + 5)(r + 5) = 80

Calculate the square root of the right side: 8.94427191

Break this problem into two subproblems by setting 
(r + 5) equal to 8.94427191 and -8.94427191.

Subproblem 1

r + 5 = 8.94427191 Simplifying r + 5 = 8.94427191 Reorder the terms: 5 + r = 8.94427191 Solving 5 + r = 8.94427191 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + r = 8.94427191 + -5 Combine like terms: 5 + -5 = 0 0 + r = 8.94427191 + -5 r = 8.94427191 + -5 Combine like terms: 8.94427191 + -5 = 3.94427191 r = 3.94427191 Simplifying r = 3.94427191

Subproblem 2

r + 5 = -8.94427191 Simplifying r + 5 = -8.94427191 Reorder the terms: 5 + r = -8.94427191 Solving 5 + r = -8.94427191 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + r = -8.94427191 + -5 Combine like terms: 5 + -5 = 0 0 + r = -8.94427191 + -5 r = -8.94427191 + -5 Combine like terms: -8.94427191 + -5 = -13.94427191 r = -13.94427191 Simplifying r = -13.94427191

Solution

The solution to the problem is based on the solutions from the subproblems. r = {3.94427191, -13.94427191}

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