Mathematics
The sum of the squares of two consecutive odd natural numbers is 394. Find the numbers.
Quadratic Equations
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Answer
Let the two consecutive odd natural numbers be x and x + 2.
It is given that the sum of the squares of the numbers = 394.
⇒ x2 + (x + 2)2 = 394
⇒ x2 + x2 + 22 + 4x = 394
⇒ 2x2 + 4 + 4x - 394 = 0
⇒ 2x2 + 4x - 390 = 0
⇒ x2 + 2x - 195 = 0
⇒ x2 + 15x - 13x - 195 = 0
⇒ x(x + 15) - 13(x + 15) = 0
⇒ (x + 15)(x - 13) = 0
⇒ (x + 15) = 0 or (x - 13) = 0
⇒ x = -15 or x = 13
As the number is natural number, x cannot be equal to -15. So, x = 13.
Other odd natural number = 13 + 2 = 15
Hence, the natural numbers are 13 and 15.
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