(r-3)(r+15)=-6

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Solution for (r-3)(r+15)=-6 equation:


Simplifying
(r + -3)(r + 15) = -6

Reorder the terms:
(-3 + r)(r + 15) = -6

Reorder the terms:
(-3 + r)(15 + r) = -6

Multiply (-3 + r) * (15 + r)
(-3(15 + r) + r(15 + r)) = -6
((15 * -3 + r * -3) + r(15 + r)) = -6
((-45 + -3r) + r(15 + r)) = -6
(-45 + -3r + (15 * r + r * r)) = -6
(-45 + -3r + (15r + r2)) = -6

Combine like terms: -3r + 15r = 12r
(-45 + 12r + r2) = -6

Solving
-45 + 12r + r2 = -6

Solving for variable 'r'.

Reorder the terms:
-45 + 6 + 12r + r2 = -6 + 6

Combine like terms: -45 + 6 = -39
-39 + 12r + r2 = -6 + 6

Combine like terms: -6 + 6 = 0
-39 + 12r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '39' to each side of the equation.
-39 + 12r + 39 + r2 = 0 + 39

Reorder the terms:
-39 + 39 + 12r + r2 = 0 + 39

Combine like terms: -39 + 39 = 0
0 + 12r + r2 = 0 + 39
12r + r2 = 0 + 39

Combine like terms: 0 + 39 = 39
12r + r2 = 39

The r term is 12r.  Take half its coefficient (6).
Square it (36) and add it to both sides.

Add '36' to each side of the equation.
12r + 36 + r2 = 39 + 36

Reorder the terms:
36 + 12r + r2 = 39 + 36

Combine like terms: 39 + 36 = 75
36 + 12r + r2 = 75

Factor a perfect square on the left side:
(r + 6)(r + 6) = 75

Calculate the square root of the right side: 8.660254038

Break this problem into two subproblems by setting 
(r + 6) equal to 8.660254038 and -8.660254038.

Subproblem 1

r + 6 = 8.660254038 Simplifying r + 6 = 8.660254038 Reorder the terms: 6 + r = 8.660254038 Solving 6 + r = 8.660254038 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = 8.660254038 + -6 Combine like terms: 6 + -6 = 0 0 + r = 8.660254038 + -6 r = 8.660254038 + -6 Combine like terms: 8.660254038 + -6 = 2.660254038 r = 2.660254038 Simplifying r = 2.660254038

Subproblem 2

r + 6 = -8.660254038 Simplifying r + 6 = -8.660254038 Reorder the terms: 6 + r = -8.660254038 Solving 6 + r = -8.660254038 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + r = -8.660254038 + -6 Combine like terms: 6 + -6 = 0 0 + r = -8.660254038 + -6 r = -8.660254038 + -6 Combine like terms: -8.660254038 + -6 = -14.660254038 r = -14.660254038 Simplifying r = -14.660254038

Solution

The solution to the problem is based on the solutions from the subproblems. r = {2.660254038, -14.660254038}

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