Problems Leading to Quadratic Equations

Example 9.3.1:

The difference of the squares of two positive integers is 325 and their difference is 5. What are the numbers?

Solution:

Let the two numbers be x and y

⇒ x – y = 5       ……….(i)

and  

⇒ x² – y² = 325        ………….(ii)

⇒  x = y + 5     ………..(iii)

Substitute for x in equation (ii)

(y + 5)² – y² = 325 

y² + 10y + 25 – y² = 325

10y + 25 = 325

10y = 300

y = \( \frac{300}{10} \)

⇒ y = 30

Substitute for y in equation (iii)

x = y + 5     ………..(iii)

x = 30 + 5

⇒  x = 35

Check:

Example 9.3.2:

The sum of the squares of two integers is 369 and their difference is 3. What are the numbers?