Question

The difference between the squares of two numbers is 45. The square of the smaller number is 4 times the larger number. Determine the numbers.

Answer 1

Let the larger number = x
then smaller number = y
Now according to the condition,
x² - y² = 45     ....(i)
and y² = 4x    ....(ii)
Substituting the value of y² from (ii) in (i)
x² - 4x = 45
⇒ x² - 4x - 45 = 0
⇒ x² - 9x + 5x - 45 = 0
⇒ x(x - 9) + 5(x - 9) = 0
⇒ (x - 9)(x + 5) = 0
Either x - 9 = 0,
then x = 9
or x + 5 = 0,
then x = -5
(i) When x = 9, the larger number = 9
and smaller number
y = `sqrt(4x)`
= `sqrt(4 xx 9)`
= `sqrt(36)`
∴ y = 6
(ii) When x = -5, then larger number = -5
y = `sqrt(4x)`
= `sqrt(4 xx 5)`
= `sqrt(-20)`
which is not possible.
Hence numbers are 6, 9.