# Thread: How to extract values from an equation

1. ## How to extract values from an equation

How do I get n from this:

I get stuck at

v/R - 1/4 = 1/n^2

2. Originally Posted by prettiestfriend
How do I get n from this:

I get stuck at

v/R - 1/4 = 1/n^2
Go ahead and do the subtraction on the left. The "common denominator" is 4R so you have 4v/(4R)- R/(4R)= (4v- R)/(4R)= 1/n^2. Now take the reciprocal of both sides: 4R/(4v- R)= n^2. Finally, just take the square root of both sides.

3. $\displaystyle v = R \bigg{(}\frac{1}{4} - \frac{1}{n^2} \bigg{)}$

$\displaystyle => \frac{v}{R} = \frac{1}{4} - \frac{1}{n^2}$

$\displaystyle => \frac{v}{R} - \frac{1}{4} = -\frac{1}{n^2}$

$\displaystyle => \frac{1}{4} - \frac{v}{R} = \frac{1}{n^2}$

Now $\displaystyle \frac{1}{4} - \frac{v}{R}= \frac{R - 4v}{4R}$ and then multiply both sides by $\displaystyle n^2$...

$\displaystyle => n^2 \cdot \frac{R - 4v}{4R} = 1$

Then divide both sides by $\displaystyle \frac{R - 4v}{4R}$...

$\displaystyle n^2 = \frac{1}{\frac{R - 4v}{4R}} = \frac{4R}{R - 4v}$

Then take the square root of each side to get...

$\displaystyle n = \sqrt{\frac{4R}{R - 4v}}$

4. Hello, prettiestfriend!

Solve for $\displaystyle n\!:\;\;v \;=\;R\left(\frac{1}{4} - \frac{1}{n^2}\right)$
Get rid of the fractions as soon as possible . . .

We have: .$\displaystyle v \;=\;\frac{R}{4} - \frac{R}{n^2}$

Multiply by $\displaystyle 4n^2\!:\quad 4n^2(v) \;=\;{\color{red}\rlap{/}}4n^2\left(\frac{R}{{\color{red}\rlap{/}}4}\right) - 4{\color{red}\rlap{//}}n^2\left(\frac{R}{{\color{red}\rlap{//}}n^2}\right)$

. . and we have: .$\displaystyle 4vn^2 \;=\;Rn^2 - 4R\quad\Rightarrow\quad Rn^2 - 4vn^2 \:=\:4R$

Factor: .$\displaystyle (R - 4v)n^2 \;=\;4R \quad\Rightarrow\quad n^2 \;=\;\frac{4R}{R - 4v}$

Take the square root: .$\displaystyle n \;=\;\pm\sqrt{\frac{4R}{R - 4v}}$