The sum of the squares of two consecutive odd positive integers is 394. Find then

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12 and 17

13 and 15

18 and 20

21 and 25

answer is B.

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Detailed Solution

Given:
Sum of squares of the numbers is 394. Also, they are consecutive odd positive numbers.
Assume the consecutive odd positive numbers as x and x + 2 and form an equation using given data.

⇒ (x) 2 + (x + 2) 2 = 394 ⇒ x 2 + x 2 + 4 x + 4 = 394 ⇒ 2 x 2 + 4 x + 4 = 394 ⇒ 2 x 2 + 4 x − 390 = 0 ⇒ 2 x 2 + 2 x − 195 = 0 ⇒ x 2 + 2 x − 195 = 0

Solve the quadratic equation using the formula to find the value of x.

x = − (2) ± (2) 2 − 4 × 1 × (− 1 2 × 1 ⇒ x = − 2 ± 4 + (4 × 195) 2 ⇒ x = − 2 ± 4 + 780 2

⇒ x = − 2 ± 784 2

⇒ x = − 2 ± 28 2

⇒ x = 2 (− 1 ± 14) 2

⇒ x = − 1 ± 14

That is, the number x can be

− 1 + 14

− 1 − 14

That is, x can be 13 or - 15.
Since - 15 is not a positive number, and 13 is an odd positive integer, 13 is the only solution.
The other number,

⇒ x + 2 ⇒ 13 + 2 ⇒ 15

Thus, the two numbers are 13 and 15 and the correct option is 2.

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