Mathematics
Find the two natural numbers which differ by 5 and the sum of whose squares is 97.
Answer
Let two natural numbers with difference 5 be x and (x - 5).
Given, sum of squares is 97.
∴ x2 + (x - 5)2 = 97
⇒ x2 + x2 + 25 - 10x = 97
⇒ 2x2 - 10x + 25 - 97 = 0
⇒ 2x2 - 10x - 72 = 0
⇒ 2(x2 - 5x - 36) = 0
⇒ x2 - 5x - 36 = 0
⇒ x2 - 9x + 4x - 36 = 0
⇒ x(x - 9) + 4(x - 9) = 0
⇒ (x + 4)(x - 9) = 0
⇒ x + 4 = 0 or x - 9 = 0
⇒ x = -4 or x = 9.
Since, x is a natural number so, x ≠ -4.
∴ x = 9 and (x - 5) = 4.
Hence, numbers = 4, 9.
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