6.5+3/2r=12+2r

Solution for 6.5+3/2r=12+2r equation:

6.5+3/2r=12+2r

We move all terms to the left:

6.5+3/2r-(12+2r)=0


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R


We add all the numbers together, and all the variables

3/2r-(2r+12)+6.5=0

We get rid of parentheses

3/2r-2r-12+6.5=0

We multiply all the terms by the denominator


-2r*2r-12*2r+(6.5)*2r+3=0

We multiply parentheses

-2r*2r-12*2r+13r+3=0

Wy multiply elements

-4r^2-24r+13r+3=0

We add all the numbers together, and all the variables

-4r^2-11r+3=0

a = -4; b = -11; c = +3;
Δ = b2-4ac
Δ = -112-4·(-4)·3
Δ = 169
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

$\sqrt{\Delta}=\sqrt{169}=13$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-11)-13}{2*-4}=\frac{-2}{-8}
=1/4
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-11)+13}{2*-4}=\frac{24}{-8}
=-3
$