(12r)^2-(5r)^2=169

Solution for (12r)^2-(5r)^2=169 equation:

(12r)^2-(5r)^2=169

We move all terms to the left:

(12r)^2-(5r)^2-(169)=0

We add all the numbers together, and all the variables

7r^2-169=0

a = 7; b = 0; c = -169;
Δ = b2-4ac
Δ = 02-4·7·(-169)
Δ = 4732
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:

$\sqrt{\Delta}=\sqrt{4732}=\sqrt{676*7}=\sqrt{676}*\sqrt{7}=26\sqrt{7}$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(0)-26\sqrt{7}}{2*7}=\frac{0-26\sqrt{7}}{14}
=-\frac{26\sqrt{7}}{14}
=-\frac{13\sqrt{7}}{7}
$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(0)+26\sqrt{7}}{2*7}=\frac{0+26\sqrt{7}}{14}
=\frac{26\sqrt{7}}{14}
=\frac{13\sqrt{7}}{7}
$