3/4n+9=n+6

Solution for 3/4n+9=n+6 equation:

3/4n+9=n+6

We move all terms to the left:

3/4n+9-(n+6)=0


Domain of the equation: 4n!=0

n!=0/4

n!=0

n∈R


We get rid of parentheses

3/4n-n-6+9=0

We multiply all the terms by the denominator


-n*4n-6*4n+9*4n+3=0

Wy multiply elements

-4n^2-24n+36n+3=0

We add all the numbers together, and all the variables

-4n^2+12n+3=0

a = -4; b = 12; c = +3;
Δ = b2-4ac
Δ = 122-4·(-4)·3
Δ = 192
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:

$\sqrt{\Delta}=\sqrt{192}=\sqrt{64*3}=\sqrt{64}*\sqrt{3}=8\sqrt{3}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(12)-8\sqrt{3}}{2*-4}=\frac{-12-8\sqrt{3}}{-8}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(12)+8\sqrt{3}}{2*-4}=\frac{-12+8\sqrt{3}}{-8}
$