4(n-5)=1/3n+2

Solution for 4(n-5)=1/3n+2 equation:

4(n-5)=1/3n+2

We move all terms to the left:

4(n-5)-(1/3n+2)=0


Domain of the equation: 3n+2)!=0

n∈R


We multiply parentheses

4n-(1/3n+2)-20=0

We get rid of parentheses

4n-1/3n-2-20=0

We multiply all the terms by the denominator


4n*3n-2*3n-20*3n-1=0

Wy multiply elements

12n^2-6n-60n-1=0

We add all the numbers together, and all the variables

12n^2-66n-1=0

a = 12; b = -66; c = -1;
Δ = b2-4ac
Δ = -662-4·12·(-1)
Δ = 4404
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:

$\sqrt{\Delta}=\sqrt{4404}=\sqrt{4*1101}=\sqrt{4}*\sqrt{1101}=2\sqrt{1101}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-66)-2\sqrt{1101}}{2*12}=\frac{66-2\sqrt{1101}}{24}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-66)+2\sqrt{1101}}{2*12}=\frac{66+2\sqrt{1101}}{24}
$