1/2r+3/2r-2=1/4r+6

Solution for 1/2r+3/2r-2=1/4r+6 equation:

1/2r+3/2r-2=1/4r+6

We move all terms to the left:

1/2r+3/2r-2-(1/4r+6)=0


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R



Domain of the equation: 4r+6)!=0

r∈R


We get rid of parentheses

1/2r+3/2r-1/4r-6-2=0

We calculate fractions


(12r+1)/8r^2+(-2r)/8r^2-6-2=0

We add all the numbers together, and all the variables

(12r+1)/8r^2+(-2r)/8r^2-8=0

We multiply all the terms by the denominator


(12r+1)+(-2r)-8*8r^2=0

Wy multiply elements

-64r^2+(12r+1)+(-2r)=0

We get rid of parentheses

-64r^2+12r-2r+1=0

We add all the numbers together, and all the variables

-64r^2+10r+1=0

a = -64; b = 10; c = +1;
Δ = b2-4ac
Δ = 102-4·(-64)·1
Δ = 356
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:

$\sqrt{\Delta}=\sqrt{356}=\sqrt{4*89}=\sqrt{4}*\sqrt{89}=2\sqrt{89}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(10)-2\sqrt{89}}{2*-64}=\frac{-10-2\sqrt{89}}{-128}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(10)+2\sqrt{89}}{2*-64}=\frac{-10+2\sqrt{89}}{-128}
$