(r-3)(r+13)=-19

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Solution for (r-3)(r+13)=-19 equation:


Simplifying
(r + -3)(r + 13) = -19

Reorder the terms:
(-3 + r)(r + 13) = -19

Reorder the terms:
(-3 + r)(13 + r) = -19

Multiply (-3 + r) * (13 + r)
(-3(13 + r) + r(13 + r)) = -19
((13 * -3 + r * -3) + r(13 + r)) = -19
((-39 + -3r) + r(13 + r)) = -19
(-39 + -3r + (13 * r + r * r)) = -19
(-39 + -3r + (13r + r2)) = -19

Combine like terms: -3r + 13r = 10r
(-39 + 10r + r2) = -19

Solving
-39 + 10r + r2 = -19

Solving for variable 'r'.

Reorder the terms:
-39 + 19 + 10r + r2 = -19 + 19

Combine like terms: -39 + 19 = -20
-20 + 10r + r2 = -19 + 19

Combine like terms: -19 + 19 = 0
-20 + 10r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '20' to each side of the equation.
-20 + 10r + 20 + r2 = 0 + 20

Reorder the terms:
-20 + 20 + 10r + r2 = 0 + 20

Combine like terms: -20 + 20 = 0
0 + 10r + r2 = 0 + 20
10r + r2 = 0 + 20

Combine like terms: 0 + 20 = 20
10r + r2 = 20

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 = 20 + 25

Reorder the terms:
25 + 10r + r2 = 20 + 25

Combine like terms: 20 + 25 = 45
25 + 10r + r2 = 45

Factor a perfect square on the left side:
(r + 5)(r + 5) = 45

Calculate the square root of the right side: 6.708203933

Break this problem into two subproblems by setting 
(r + 5) equal to 6.708203933 and -6.708203933.

Subproblem 1

r + 5 = 6.708203933 Simplifying r + 5 = 6.708203933 Reorder the terms: 5 + r = 6.708203933 Solving 5 + r = 6.708203933 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 = 6.708203933 + -5 Combine like terms: 5 + -5 = 0 0 + r = 6.708203933 + -5 r = 6.708203933 + -5 Combine like terms: 6.708203933 + -5 = 1.708203933 r = 1.708203933 Simplifying r = 1.708203933

Subproblem 2

r + 5 = -6.708203933 Simplifying r + 5 = -6.708203933 Reorder the terms: 5 + r = -6.708203933 Solving 5 + r = -6.708203933 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 = -6.708203933 + -5 Combine like terms: 5 + -5 = 0 0 + r = -6.708203933 + -5 r = -6.708203933 + -5 Combine like terms: -6.708203933 + -5 = -11.708203933 r = -11.708203933 Simplifying r = -11.708203933

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.708203933, -11.708203933}

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