1/2r+3=2r-1

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

1/2r+3=2r-1

We move all terms to the left:

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


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R


We get rid of parentheses

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

We multiply all the terms by the denominator


-2r*2r+1*2r+3*2r+1=0

Wy multiply elements

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

We add all the numbers together, and all the variables

-4r^2+8r+1=0

a = -4; b = 8; c = +1;
Δ = b2-4ac
Δ = 82-4·(-4)·1
Δ = 80
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:

$\sqrt{\Delta}=\sqrt{80}=\sqrt{16*5}=\sqrt{16}*\sqrt{5}=4\sqrt{5}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(8)-4\sqrt{5}}{2*-4}=\frac{-8-4\sqrt{5}}{-8}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(8)+4\sqrt{5}}{2*-4}=\frac{-8+4\sqrt{5}}{-8}
$