5n^2-42=1

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

5n^2-42=1

We move all terms to the left:

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

We add all the numbers together, and all the variables

5n^2-43=0

a = 5; b = 0; c = -43;
Δ = b2-4ac
Δ = 02-4·5·(-43)
Δ = 860
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:

$\sqrt{\Delta}=\sqrt{860}=\sqrt{4*215}=\sqrt{4}*\sqrt{215}=2\sqrt{215}$
$n_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-2\sqrt{215}}{2*5}=\frac{0-2\sqrt{215}}{10}
=-\frac{2\sqrt{215}}{10}
=-\frac{\sqrt{215}}{5}
$
$n_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+2\sqrt{215}}{2*5}=\frac{0+2\sqrt{215}}{10}
=\frac{2\sqrt{215}}{10}
=\frac{\sqrt{215}}{5}
$