24r^2-5r-6=0

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Solution for 24r^2-5r-6=0 equation: 


Simplifying
24r2 + -5r + -6 = 0

Reorder the terms:
-6 + -5r + 24r2 = 0

Solving
-6 + -5r + 24r2 = 0

Solving for variable 'r'.

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

Divide each side by '24'.
-0.25 + -0.2083333333r + r2 = 0

Move the constant term to the right:

Add '0.25' to each side of the equation.
-0.25 + -0.2083333333r + 0.25 + r2 = 0 + 0.25

Reorder the terms:
-0.25 + 0.25 + -0.2083333333r + r2 = 0 + 0.25

Combine like terms: -0.25 + 0.25 = 0.00
0.00 + -0.2083333333r + r2 = 0 + 0.25
-0.2083333333r + r2 = 0 + 0.25

Combine like terms: 0 + 0.25 = 0.25
-0.2083333333r + r2 = 0.25

The r term is -0.2083333333r.  Take half its coefficient (-0.1041666667).
Square it (0.01085069445) and add it to both sides.

Add '0.01085069445' to each side of the equation.
-0.2083333333r + 0.01085069445 + r2 = 0.25 + 0.01085069445

Reorder the terms:
0.01085069445 + -0.2083333333r + r2 = 0.25 + 0.01085069445

Combine like terms: 0.25 + 0.01085069445 = 0.26085069445
0.01085069445 + -0.2083333333r + r2 = 0.26085069445

Factor a perfect square on the left side:
(r + -0.1041666667)(r + -0.1041666667) = 0.26085069445

Calculate the square root of the right side: 0.510735445

Break this problem into two subproblems by setting 
(r + -0.1041666667) equal to 0.510735445 and -0.510735445.


Subproblem 1
r + -0.1041666667 = 0.510735445

Simplifying
r + -0.1041666667 = 0.510735445

Reorder the terms:
-0.1041666667 + r = 0.510735445

Solving
-0.1041666667 + r = 0.510735445

Solving for variable 'r'.

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

Add '0.1041666667' to each side of the equation.
-0.1041666667 + 0.1041666667 + r = 0.510735445 + 0.1041666667

Combine like terms: -0.1041666667 + 0.1041666667 = 0.0000000000
0.0000000000 + r = 0.510735445 + 0.1041666667
r = 0.510735445 + 0.1041666667

Combine like terms: 0.510735445 + 0.1041666667 = 0.6149021117
r = 0.6149021117

Simplifying
r = 0.6149021117


Subproblem 2
r + -0.1041666667 = -0.510735445

Simplifying
r + -0.1041666667 = -0.510735445

Reorder the terms:
-0.1041666667 + r = -0.510735445

Solving
-0.1041666667 + r = -0.510735445

Solving for variable 'r'.

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

Add '0.1041666667' to each side of the equation.
-0.1041666667 + 0.1041666667 + r = -0.510735445 + 0.1041666667

Combine like terms: -0.1041666667 + 0.1041666667 = 0.0000000000
0.0000000000 + r = -0.510735445 + 0.1041666667
r = -0.510735445 + 0.1041666667

Combine like terms: -0.510735445 + 0.1041666667 = -0.4065687783
r = -0.4065687783

Simplifying
r = -0.4065687783


Solution
The solution to the problem is based on the solutions
from the subproblems.
r = {0.6149021117, -0.4065687783}

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