(r-3)(r+9)=-16

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Solution for (r-3)(r+9)=-16 equation:


Simplifying
(r + -3)(r + 9) = -16

Reorder the terms:
(-3 + r)(r + 9) = -16

Reorder the terms:
(-3 + r)(9 + r) = -16

Multiply (-3 + r) * (9 + r)
(-3(9 + r) + r(9 + r)) = -16
((9 * -3 + r * -3) + r(9 + r)) = -16
((-27 + -3r) + r(9 + r)) = -16
(-27 + -3r + (9 * r + r * r)) = -16
(-27 + -3r + (9r + r2)) = -16

Combine like terms: -3r + 9r = 6r
(-27 + 6r + r2) = -16

Solving
-27 + 6r + r2 = -16

Solving for variable 'r'.

Reorder the terms:
-27 + 16 + 6r + r2 = -16 + 16

Combine like terms: -27 + 16 = -11
-11 + 6r + r2 = -16 + 16

Combine like terms: -16 + 16 = 0
-11 + 6r + r2 = 0

Begin completing the square.

Move the constant term to the right:

Add '11' to each side of the equation.
-11 + 6r + 11 + r2 = 0 + 11

Reorder the terms:
-11 + 11 + 6r + r2 = 0 + 11

Combine like terms: -11 + 11 = 0
0 + 6r + r2 = 0 + 11
6r + r2 = 0 + 11

Combine like terms: 0 + 11 = 11
6r + r2 = 11

The r term is 6r.  Take half its coefficient (3).
Square it (9) and add it to both sides.

Add '9' to each side of the equation.
6r + 9 + r2 = 11 + 9

Reorder the terms:
9 + 6r + r2 = 11 + 9

Combine like terms: 11 + 9 = 20
9 + 6r + r2 = 20

Factor a perfect square on the left side:
(r + 3)(r + 3) = 20

Calculate the square root of the right side: 4.472135955

Break this problem into two subproblems by setting 
(r + 3) equal to 4.472135955 and -4.472135955.

Subproblem 1

r + 3 = 4.472135955 Simplifying r + 3 = 4.472135955 Reorder the terms: 3 + r = 4.472135955 Solving 3 + r = 4.472135955 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = 4.472135955 + -3 Combine like terms: 3 + -3 = 0 0 + r = 4.472135955 + -3 r = 4.472135955 + -3 Combine like terms: 4.472135955 + -3 = 1.472135955 r = 1.472135955 Simplifying r = 1.472135955

Subproblem 2

r + 3 = -4.472135955 Simplifying r + 3 = -4.472135955 Reorder the terms: 3 + r = -4.472135955 Solving 3 + r = -4.472135955 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + r = -4.472135955 + -3 Combine like terms: 3 + -3 = 0 0 + r = -4.472135955 + -3 r = -4.472135955 + -3 Combine like terms: -4.472135955 + -3 = -7.472135955 r = -7.472135955 Simplifying r = -7.472135955

Solution

The solution to the problem is based on the solutions from the subproblems. r = {1.472135955, -7.472135955}

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