6(1+n)=8n(n-7)

Solution for 6(1+n)=8n(n-7) equation:

6(1+n)=8n(n-7)

We move all terms to the left:

6(1+n)-(8n(n-7))=0

We add all the numbers together, and all the variables

6(n+1)-(8n(n-7))=0

We multiply parentheses

6n-(8n(n-7))+6=0


We calculate terms in parentheses: -(8n(n-7)), so:

8n(n-7)

We multiply parentheses

8n^2-56n

Back to the equation:

-(8n^2-56n)


We get rid of parentheses

-8n^2+6n+56n+6=0

We add all the numbers together, and all the variables

-8n^2+62n+6=0

a = -8; b = 62; c = +6;
Δ = b2-4ac
Δ = 622-4·(-8)·6
Δ = 4036
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:

$\sqrt{\Delta}=\sqrt{4036}=\sqrt{4*1009}=\sqrt{4}*\sqrt{1009}=2\sqrt{1009}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(62)-2\sqrt{1009}}{2*-8}=\frac{-62-2\sqrt{1009}}{-16}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(62)+2\sqrt{1009}}{2*-8}=\frac{-62+2\sqrt{1009}}{-16}
$