(5n-17)(2n+1)=n

Solution for (5n-17)(2n+1)=n equation:

(5n-17)(2n+1)=n

We move all terms to the left:

(5n-17)(2n+1)-(n)=0

We add all the numbers together, and all the variables

-1n+(5n-17)(2n+1)=0

We multiply parentheses ..

(+10n^2+5n-34n-17)-1n=0

We get rid of parentheses

10n^2+5n-34n-1n-17=0

We add all the numbers together, and all the variables

10n^2-30n-17=0

a = 10; b = -30; c = -17;
Δ = b2-4ac
Δ = -302-4·10·(-17)
Δ = 1580
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:

$\sqrt{\Delta}=\sqrt{1580}=\sqrt{4*395}=\sqrt{4}*\sqrt{395}=2\sqrt{395}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-30)-2\sqrt{395}}{2*10}=\frac{30-2\sqrt{395}}{20}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-30)+2\sqrt{395}}{2*10}=\frac{30+2\sqrt{395}}{20}
$