0.5r-3=12-9/2r

Solution for 0.5r-3=12-9/2r equation:

0.5r-3=12-9/2r

We move all terms to the left:

0.5r-3-(12-9/2r)=0


Domain of the equation: 2r)!=0

r!=0/1

r!=0

r∈R


We add all the numbers together, and all the variables

0.5r-(-9/2r+12)-3=0

We get rid of parentheses

0.5r+9/2r-12-3=0

We multiply all the terms by the denominator


(0.5r)*2r-12*2r-3*2r+9=0

We add all the numbers together, and all the variables

(+0.5r)*2r-12*2r-3*2r+9=0

We multiply parentheses

0r^2-12*2r-3*2r+9=0

Wy multiply elements

0r^2-24r-6r+9=0

We add all the numbers together, and all the variables

r^2-30r+9=0

a = 1; b = -30; c = +9;
Δ = b2-4ac
Δ = -302-4·1·9
Δ = 864
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:

$\sqrt{\Delta}=\sqrt{864}=\sqrt{144*6}=\sqrt{144}*\sqrt{6}=12\sqrt{6}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-12\sqrt{6}}{2*1}=\frac{30-12\sqrt{6}}{2}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+12\sqrt{6}}{2*1}=\frac{30+12\sqrt{6}}{2}
$