Question

The difference of square of two numbers is 45 and square of smaller numberis four times the larger number .find the two numbers​

Answer 1

Answer:

$\large{\underline{\boxed{\sf 9 \: \: and \: \: 6}}}$

Step-by-step explanation:

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]

_______________________

Therefore, the required numbers are-

• x = 9
• y = 6

Hence, the numbers are 9 and 6.

Answer 2

Answer:

Step-by-step explanation:

Smaller number (y) in terms of larger number (x):  

x² - y² = 45  

y² = x² - 45  

Value of larger number (x):  

x² - 45 = 4x  

x² - 2x = 45 + (- 2)²  

x² - 2x = 45 + 4  

(x - 2)² = 49  

x - 2 = 7  

x = 9  

Value of smaller number:  

= √(9² - 45)  

= √(81 - 45)  

= √36  

= 6  

Hence, the two numbers​ are 9 and 6.

Verification :-

Proof (difference of their squares is 45):  

= 9² - 6²  

= 81 - 36  

= 45  

Proof (square of smaller is equal to 4 times the larger):  

6² = 4(9)  

36 = 36