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Sum of the squares of two natural numbers is 97. If numbers differ by 5, then what is the positive difference in their squares?

Sum of the squares of two natural numbers is 97. If numbers differ by 5, then what is the positive difference in their squares? Correct Answer 65

Given:

Sum of the squares of two natural numbers = 97

Difference of the number = 5

Calculation:

Let the two natural numbers are x and y

So,x2 + y2 = 97      ---- (1)

x - y = 5

⇒ x = 5 + y      ---- (2)

(5 + y)2 + y2 = 97  (From 1 and 2)

⇒ 25 + y2 + 10y + y2 = 97

⇒ 2y2 + 10y = 97 - 25

⇒ 2y2 + 10y - 72 = 0

⇒ y2 + 5y - 36 = 0

⇒ y2 + 9y - 4y - 36 = 0

⇒ y(y + 9) - 4(y + 9) = 0

⇒ (y - 4) (y + 9) = 0

⇒ y = 4, y = -9

x = 5 + 4 (from 2)

⇒ 9

Required number = (9)2 - (4)2

⇒ 81 - 16

⇒ 65

∴ The positive difference in their squares is 65.

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