Let us say the larger and smaller number be x and y, respectively.
As per the question given,
$x^{2} – y^{2} $= 180 and $y^{2}$ = 8x
⇒ $x^{2} – 8x$ = 180
⇒ $x^{2} – 8x – 180$ = 0
⇒ $x^{2} – 18x + 10x – 180$ = 0
⇒ x(x – 18) +10(x – 18) = 0
⇒ (x – 18)(x + 10) = 0
⇒ x = 18, -10
The larger number will be 18 only.
x = 18
∴ $y^{2}$ = 8x = 8 × 18 = 144
⇒ y =$ \pm\sqrt{144}$ = ±12
∴ Smaller number = ±12
Therefore, the numbers are 18 and 12 or 18 and -12.