1/3r-2=9r+15

Solution for 1/3r-2=9r+15 equation:

1/3r-2=9r+15

We move all terms to the left:

1/3r-2-(9r+15)=0


Domain of the equation: 3r!=0

r!=0/3

r!=0

r∈R


We get rid of parentheses

1/3r-9r-15-2=0

We multiply all the terms by the denominator


-9r*3r-15*3r-2*3r+1=0

Wy multiply elements

-27r^2-45r-6r+1=0

We add all the numbers together, and all the variables

-27r^2-51r+1=0

a = -27; b = -51; c = +1;
Δ = b2-4ac
Δ = -512-4·(-27)·1
Δ = 2709
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{2709}=\sqrt{9*301}=\sqrt{9}*\sqrt{301}=3\sqrt{301}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-51)-3\sqrt{301}}{2*-27}=\frac{51-3\sqrt{301}}{-54}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-51)+3\sqrt{301}}{2*-27}=\frac{51+3\sqrt{301}}{-54}
$