9r^2+2r-24=0

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Solution for 9r^2+2r-24=0 equation: 


Simplifying
9r2 + 2r + -24 = 0

Reorder the terms:
-24 + 2r + 9r2 = 0

Solving
-24 + 2r + 9r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
9 the coefficient of the squared term: 

Divide each side by '9'.
-2.666666667 + 0.2222222222r + r2 = 0

Move the constant term to the right:

Add '2.666666667' to each side of the equation.
-2.666666667 + 0.2222222222r + 2.666666667 + r2 = 0 + 2.666666667

Reorder the terms:
-2.666666667 + 2.666666667 + 0.2222222222r + r2 = 0 + 2.666666667

Combine like terms: -2.666666667 + 2.666666667 = 0.000000000
0.000000000 + 0.2222222222r + r2 = 0 + 2.666666667
0.2222222222r + r2 = 0 + 2.666666667

Combine like terms: 0 + 2.666666667 = 2.666666667
0.2222222222r + r2 = 2.666666667

The r term is 0.2222222222r.  Take half its coefficient (0.1111111111).
Square it (0.01234567901) and add it to both sides.

Add '0.01234567901' to each side of the equation.
0.2222222222r + 0.01234567901 + r2 = 2.666666667 + 0.01234567901

Reorder the terms:
0.01234567901 + 0.2222222222r + r2 = 2.666666667 + 0.01234567901

Combine like terms: 2.666666667 + 0.01234567901 = 2.67901234601
0.01234567901 + 0.2222222222r + r2 = 2.67901234601

Factor a perfect square on the left side:
(r + 0.1111111111)(r + 0.1111111111) = 2.67901234601

Calculate the square root of the right side: 1.636768874

Break this problem into two subproblems by setting 
(r + 0.1111111111) equal to 1.636768874 and -1.636768874.


Subproblem 1
r + 0.1111111111 = 1.636768874

Simplifying
r + 0.1111111111 = 1.636768874

Reorder the terms:
0.1111111111 + r = 1.636768874

Solving
0.1111111111 + r = 1.636768874

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -0.1111111111 + r = 1.636768874 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + r = 1.636768874 + -0.1111111111
r = 1.636768874 + -0.1111111111

Combine like terms: 1.636768874 + -0.1111111111 = 1.5256577629
r = 1.5256577629

Simplifying
r = 1.5256577629


Subproblem 2
r + 0.1111111111 = -1.636768874

Simplifying
r + 0.1111111111 = -1.636768874

Reorder the terms:
0.1111111111 + r = -1.636768874

Solving
0.1111111111 + r = -1.636768874

Solving for variable 'r'.

Move all terms containing r to the left, all other terms to the right.

Add '-0.1111111111' to each side of the equation.
0.1111111111 + -0.1111111111 + r = -1.636768874 + -0.1111111111

Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000
0.0000000000 + r = -1.636768874 + -0.1111111111
r = -1.636768874 + -0.1111111111

Combine like terms: -1.636768874 + -0.1111111111 = -1.7478799851
r = -1.7478799851

Simplifying
r = -1.7478799851


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {1.5256577629, -1.7478799851}

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