(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 = -23
-23 + 10r + r2 = -16 + 16

Combine like terms: -16 + 16 = 0
-23 + 10r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '23' to each side of the equation.
-23 + 10r + 23 + r2 = 0 + 23

Reorder the terms:
-23 + 23 + 10r + r2 = 0 + 23

Combine like terms: -23 + 23 = 0
0 + 10r + r2 = 0 + 23
10r + r2 = 0 + 23

Combine like terms: 0 + 23 = 23
10r + r2 = 23

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 = 23 + 25

Reorder the terms:
25 + 10r + r2 = 23 + 25

Combine like terms: 23 + 25 = 48
25 + 10r + r2 = 48

Factor a perfect square on the left side:
(r + 5)(r + 5) = 48

Calculate the square root of the right side: 6.92820323

Break this problem into two subproblems by setting 
(r + 5) equal to 6.92820323 and -6.92820323.

Subproblem 1

r + 5 = 6.92820323 Simplifying r + 5 = 6.92820323 Reorder the terms: 5 + r = 6.92820323 Solving 5 + r = 6.92820323 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 = 6.92820323 + -5 Combine like terms: 5 + -5 = 0 0 + r = 6.92820323 + -5 r = 6.92820323 + -5 Combine like terms: 6.92820323 + -5 = 1.92820323 r = 1.92820323 Simplifying r = 1.92820323

Subproblem 2

r + 5 = -6.92820323 Simplifying r + 5 = -6.92820323 Reorder the terms: 5 + r = -6.92820323 Solving 5 + r = -6.92820323 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 = -6.92820323 + -5 Combine like terms: 5 + -5 = 0 0 + r = -6.92820323 + -5 r = -6.92820323 + -5 Combine like terms: -6.92820323 + -5 = -11.92820323 r = -11.92820323 Simplifying r = -11.92820323

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.92820323, -11.92820323}

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