3/4n=n-5

Solution for 3/4n=n-5 equation:

3/4n=n-5

We move all terms to the left:

3/4n-(n-5)=0


Domain of the equation: 4n!=0

n!=0/4

n!=0

n∈R


We get rid of parentheses

3/4n-n+5=0

We multiply all the terms by the denominator


-n*4n+5*4n+3=0

Wy multiply elements

-4n^2+20n+3=0

a = -4; b = 20; c = +3;
Δ = b2-4ac
Δ = 202-4·(-4)·3
Δ = 448
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{448}=\sqrt{64*7}=\sqrt{64}*\sqrt{7}=8\sqrt{7}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(20)-8\sqrt{7}}{2*-4}=\frac{-20-8\sqrt{7}}{-8}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(20)+8\sqrt{7}}{2*-4}=\frac{-20+8\sqrt{7}}{-8}
$