The difference of squares of two numbers is 45 and the square of the smaller number is four times the larger number. Find the two numbers.

Let larger number be x and smaller number be y.

Given:

(Large Number)² – (Smaller Number)2 = 45

x² – y² = 45

y² = x² – 45

Also Smaller Number², y² = 4 × Large Number

x² – 45 = 4x

x² – 4x – 45 = 0

On factorizing the above equation,

Sum = -4

Product = -45

Therefore the two numbers satisfying the above conditions are -9 and 5.

x² – 9x + 5x – 45 = 0

x(x – 9) + 5(x – 9) = 0

(x – 9) (x + 5) = 0

Solving first part,

x – 9 = 0

x = 9

Solving second part,

x + 5 = 0

x = -5

If x = 9, then

y² = 4 × 9

y² = 36

y =± 6

If x = -5, then

y² = 4 × -5

y² = -20

Value of y becomes imaginary which is not possible.

So possible values of x and y are 9, 6 or 9, -6.