-9(r)=2/3r-6

Solution for -9(r)=2/3r-6 equation:

-9(r)=2/3r-6

We move all terms to the left:

-9(r)-(2/3r-6)=0


Domain of the equation: 3r-6)!=0

r∈R


We get rid of parentheses

-9r-2/3r+6=0

We multiply all the terms by the denominator


-9r*3r+6*3r-2=0

Wy multiply elements

-27r^2+18r-2=0

a = -27; b = 18; c = -2;
Δ = b2-4ac
Δ = 182-4·(-27)·(-2)
Δ = 108
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:

$\sqrt{\Delta}=\sqrt{108}=\sqrt{36*3}=\sqrt{36}*\sqrt{3}=6\sqrt{3}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(18)-6\sqrt{3}}{2*-27}=\frac{-18-6\sqrt{3}}{-54}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(18)+6\sqrt{3}}{2*-27}=\frac{-18+6\sqrt{3}}{-54}
$