7r^2+1=414

Solution for 7r^2+1=414 equation:

7r^2+1=414

We move all terms to the left:

7r^2+1-(414)=0

We add all the numbers together, and all the variables

7r^2-413=0

a = 7; b = 0; c = -413;
Δ = b2-4ac
Δ = 02-4·7·(-413)
Δ = 11564
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{11564}=\sqrt{196*59}=\sqrt{196}*\sqrt{59}=14\sqrt{59}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-14\sqrt{59}}{2*7}=\frac{0-14\sqrt{59}}{14}
=-\frac{14\sqrt{59}}{14}
=-\sqrt{59}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+14\sqrt{59}}{2*7}=\frac{0+14\sqrt{59}}{14}
=\frac{14\sqrt{59}}{14}
=\sqrt{59}
$