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

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

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

We move all terms to the left:

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


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R



Domain of the equation: 4r!=0

r!=0/4

r!=0

r∈R



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

r∈R


We get rid of parentheses

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

We calculate fractions


4r/8r^2+(-2r+6)/8r^2-6-2=0

We add all the numbers together, and all the variables

4r/8r^2+(-2r+6)/8r^2-8=0

We multiply all the terms by the denominator


4r+(-2r+6)-8*8r^2=0

Wy multiply elements

-64r^2+4r+(-2r+6)=0

We get rid of parentheses

-64r^2+4r-2r+6=0

We add all the numbers together, and all the variables

-64r^2+2r+6=0

a = -64; b = 2; c = +6;
Δ = b2-4ac
Δ = 22-4·(-64)·6
Δ = 1540
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:

$\sqrt{\Delta}=\sqrt{1540}=\sqrt{4*385}=\sqrt{4}*\sqrt{385}=2\sqrt{385}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(2)-2\sqrt{385}}{2*-64}=\frac{-2-2\sqrt{385}}{-128}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(2)+2\sqrt{385}}{2*-64}=\frac{-2+2\sqrt{385}}{-128}
$