Question

The difference of squares of two natural numbers is 45 . The square of the smaller number is 4 times the larger number.Then the numbers are ?

Answer 1

Check the attachment

Answer 2

Solution:

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

It is given that,

The difference of square of two numbers is 45.

⇒ x² - y² = 45 .... (i)

Also, square of smaller number is four times the larger number.

⇒ y² = 4x

According to the question ;

x² - y² = 45

⇒ x² - 4x = 45 [Since, y² = 4x]

⇒ x² - 4x - 45 = 0

On splitting the middle term,

⇒ x² - (9 - 5)x - 45 = 0

⇒ x² - 9x + 5x - 45 = 0

⇒ x(x - 9) + 5(x - 9) = 0

⇒ (x - 9)(x + 5) = 0

⇒ x = 9 or x = - 5

Thus, by neglecting the negative value

We get the value of x = 9.

Substituting the value of x in (i)

⇒ x² - y² = 45

⇒ (9)² - y² = 45

⇒ 81 - y² = 45

⇒ y² = 81 - 45

⇒ y² = 36

⇒ y = ± √36

⇒ y = 6 [Neglecting negative value]

$\rule{300}{2}$

Therefore, the required numbers are-

• x = 9
• y = 6

Hence, the numbers are 9 and 6.