Question

the difference between the squareS of 2 natural numbers is 45. the square of THE smaller numbers is 4 times the larger number. find the numbers

Answer 1

Given:

• Different between square of 2 natural numbers is 45
• Square of smaller number is 4 times larger number.

To Find:

• What are numbers ?

Solution:

Given that square of two natural numbers is 45

Supposing that :

• Smaller number is x
• Larger number is y

According to question:

• y² - x² = 45 ••••••(i)
• x² = 4y ••••••(ii)

In $\small{\bf{2^{nd}}}$ equation, we have mentioned that x² = 4y. So we can put value of x² in equation $\small{\bf{1^{st}}}$

Hence,

y² - x² = 45

→ y² - 4y = 45

→ y² - 4y - 45 = 0

→ y² + 5y - 9y - 45 = 0

→ y(y + 5) -9(y + 5) = 0

→ (y - 9) (y + 5)

From here, we can see two values of y i.e y = 9 and y = -5.

But, we need to ignore negative value of y.

Therefore,

• value of y = 9

Putting value of y in equation (i) :-

y² - x² = 45

→ 9² - x² = 45

→ 81 - x² = 45

→ -x² = 45 - 81

→ -x² = -36

→ x² = 36

→ x = √36

→ x = 6

Therefore,

• Required numbers are 9 and 6.

Verification :

To verify value of x and y, put values in equation (i) -

y² - x² = 45

→9² - 6² = 45

→ 81 - 36 = 45

→ 45 = 45

LHS = RHS ••••••••(Verified)

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