The difference between two numbers is 14 and the difference between their squares is 448. Find the numbers.

Let the larger number be x and the smaller number be y. 

Then, we have: 

x – y = 14 or x = 14 + y ……….(i) 

x² – y² = 448 ………(ii) 

On substituting x = 14 + y in (ii) we get 

(14 + y)² – y² = 448 

⇒ 196 + y² + 28y – y² = 448 

⇒ 196 + 28y = 448 

⇒ 28y = (448 – 196) = 252 

⇒ y = 252/28 = 9 

On substituting y = 9 in (i), we get:

 x = 14 + 9 = 23 

Hence, the required numbers are 23 and 9

Let us consider,

First number = x and

Second number = y

(Assume y is smaller)

As per the statement,

x – y = 14 …..(1)

and x² – y² = 448

(x + y) (x – y) = 448

(x + y) x 14 = 448

x + y = 32 ……(2)

Adding (1) and (2),

2x = 46 or x = 23

Subtracting (1) from (2)

2y = 18 or y = 9

Answer:

Numbers are 23 and 9

Let the larger number be x and the smaller number be y. 

Then, we have: 

x – y = 14 or x = 14 + y ……….(i) 

x² – y² = 448 ………(ii) 

On substituting x = 14 + y in (ii) we get 

(14 + y)² – y² = 448 

⇒ 196 + y² + 28y – y² = 448 

⇒ 196 + 28y = 448 

⇒ 28y = (448 – 196) = 252 

⇒ y = 252/28 = 9 

On substituting y = 9 in (i), we get: 

x = 14 + 9 = 23 

Hence, the required numbers are 23 and 9.