Question

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find two numbers.

Answer 1

Let the greater number be x and smaller number be y. Then, according to given information, we have

x² – y² = 180  ...(i)

And y² = 8x  ...(ii)

From equations (i) and (ii), we get

x² – 8x – 180 = 0

⇒ x² – 18x + 10x – 180 = 0

⇒ x(x – 18) + 10(x – 18) = 0

⇒ (x – 18)(x + 10) = 0

⇒ x = 18  ...[∵ x ≠ – 10, if x = –10, then y² = –80 not possible]

Now, from equation (ii),

y² = 8 × 18 = 144

⇒ y = ± 12

 So, the two numbers are 18 and 12 or 18 and –12.