-31+52x^2-1=0

Solution for -31+52x^2-1=0 equation:

-31+52x^2-1=0

We add all the numbers together, and all the variables

52x^2-32=0

a = 52; b = 0; c = -32;
Δ = b2-4ac
Δ = 02-4·52·(-32)
Δ = 6656
The delta value is higher than zero, so the equation has two solutions
We use following formulas to calculate our solutions:
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}$

The end solution:

$\sqrt{\Delta}=\sqrt{6656}=\sqrt{256*26}=\sqrt{256}*\sqrt{26}=16\sqrt{26}$
$x_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-16\sqrt{26}}{2*52}=\frac{0-16\sqrt{26}}{104}
=-\frac{16\sqrt{26}}{104}
=-\frac{2\sqrt{26}}{13}
$
$x_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+16\sqrt{26}}{2*52}=\frac{0+16\sqrt{26}}{104}
=\frac{16\sqrt{26}}{104}
=\frac{2\sqrt{26}}{13}
$