31+49n=1-7n^2-41n

Solution for 31+49n=1-7n^2-41n equation:

31+49n=1-7n^2-41n

We move all terms to the left:

31+49n-(1-7n^2-41n)=0

We get rid of parentheses

7n^2+41n+49n-1+31=0

We add all the numbers together, and all the variables

7n^2+90n+30=0

a = 7; b = 90; c = +30;
Δ = b2-4ac
Δ = 902-4·7·30
Δ = 7260
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:

$\sqrt{\Delta}=\sqrt{7260}=\sqrt{484*15}=\sqrt{484}*\sqrt{15}=22\sqrt{15}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(90)-22\sqrt{15}}{2*7}=\frac{-90-22\sqrt{15}}{14}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(90)+22\sqrt{15}}{2*7}=\frac{-90+22\sqrt{15}}{14}
$