10+3n=3/4n-8

Solution for 10+3n=3/4n-8 equation:

10+3n=3/4n-8

We move all terms to the left:

10+3n-(3/4n-8)=0


Domain of the equation: 4n-8)!=0

n∈R


We get rid of parentheses

3n-3/4n+8+10=0

We multiply all the terms by the denominator


3n*4n+8*4n+10*4n-3=0

Wy multiply elements

12n^2+32n+40n-3=0

We add all the numbers together, and all the variables

12n^2+72n-3=0

a = 12; b = 72; c = -3;
Δ = b2-4ac
Δ = 722-4·12·(-3)
Δ = 5328
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:

$\sqrt{\Delta}=\sqrt{5328}=\sqrt{144*37}=\sqrt{144}*\sqrt{37}=12\sqrt{37}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(72)-12\sqrt{37}}{2*12}=\frac{-72-12\sqrt{37}}{24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(72)+12\sqrt{37}}{2*12}=\frac{-72+12\sqrt{37}}{24}
$