5n^2+42n+6=-1

Solution for 5n^2+42n+6=-1 equation:

5n^2+42n+6=-1

We move all terms to the left:

5n^2+42n+6-(-1)=0

We add all the numbers together, and all the variables

5n^2+42n+7=0

a = 5; b = 42; c = +7;
Δ = b2-4ac
Δ = 422-4·5·7
Δ = 1624
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:

$\sqrt{\Delta}=\sqrt{1624}=\sqrt{4*406}=\sqrt{4}*\sqrt{406}=2\sqrt{406}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(42)-2\sqrt{406}}{2*5}=\frac{-42-2\sqrt{406}}{10}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(42)+2\sqrt{406}}{2*5}=\frac{-42+2\sqrt{406}}{10}
$