r^2-9r-27=0

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Solution for r^2-9r-27=0 equation:


Simplifying
r2 + -9r + -27 = 0

Reorder the terms:
-27 + -9r + r2 = 0

Solving
-27 + -9r + r2 = 0

Solving for variable 'r'.

Begin completing the square.

Move the constant term to the right:

Add '27' to each side of the equation.
-27 + -9r + 27 + r2 = 0 + 27

Reorder the terms:
-27 + 27 + -9r + r2 = 0 + 27

Combine like terms: -27 + 27 = 0
0 + -9r + r2 = 0 + 27
-9r + r2 = 0 + 27

Combine like terms: 0 + 27 = 27
-9r + r2 = 27

The r term is -9r.  Take half its coefficient (-4.5).
Square it (20.25) and add it to both sides.

Add '20.25' to each side of the equation.
-9r + 20.25 + r2 = 27 + 20.25

Reorder the terms:
20.25 + -9r + r2 = 27 + 20.25

Combine like terms: 27 + 20.25 = 47.25
20.25 + -9r + r2 = 47.25

Factor a perfect square on the left side:
(r + -4.5)(r + -4.5) = 47.25

Calculate the square root of the right side: 6.873863542

Break this problem into two subproblems by setting 
(r + -4.5) equal to 6.873863542 and -6.873863542.

Subproblem 1

r + -4.5 = 6.873863542 Simplifying r + -4.5 = 6.873863542 Reorder the terms: -4.5 + r = 6.873863542 Solving -4.5 + r = 6.873863542 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + r = 6.873863542 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = 6.873863542 + 4.5 r = 6.873863542 + 4.5 Combine like terms: 6.873863542 + 4.5 = 11.373863542 r = 11.373863542 Simplifying r = 11.373863542

Subproblem 2

r + -4.5 = -6.873863542 Simplifying r + -4.5 = -6.873863542 Reorder the terms: -4.5 + r = -6.873863542 Solving -4.5 + r = -6.873863542 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + r = -6.873863542 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + r = -6.873863542 + 4.5 r = -6.873863542 + 4.5 Combine like terms: -6.873863542 + 4.5 = -2.373863542 r = -2.373863542 Simplifying r = -2.373863542

Solution

The solution to the problem is based on the solutions from the subproblems. r = {11.373863542, -2.373863542}

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