1/2r+1=7r-1

Solution for 1/2r+1=7r-1 equation:

1/2r+1=7r-1

We move all terms to the left:

1/2r+1-(7r-1)=0


Domain of the equation: 2r!=0

r!=0/2

r!=0

r∈R


We get rid of parentheses

1/2r-7r+1+1=0

We multiply all the terms by the denominator


-7r*2r+1*2r+1*2r+1=0

Wy multiply elements

-14r^2+2r+2r+1=0

We add all the numbers together, and all the variables

-14r^2+4r+1=0

a = -14; b = 4; c = +1;
Δ = b2-4ac
Δ = 42-4·(-14)·1
Δ = 72
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:

$\sqrt{\Delta}=\sqrt{72}=\sqrt{36*2}=\sqrt{36}*\sqrt{2}=6\sqrt{2}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(4)-6\sqrt{2}}{2*-14}=\frac{-4-6\sqrt{2}}{-28}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(4)+6\sqrt{2}}{2*-14}=\frac{-4+6\sqrt{2}}{-28}
$