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Question:

The quadratic equation x² + bx + c = 0 has two roots 4a and 3a, where a is an integer. Which of the following is a possible value of b² + c?

Explanation:

Given quadratic equation is x² + bx + c = 0

Sum of roots = -b

⇒ -b = 4a + 3a

⇒ -b = 7a

Product of the roots = c

⇒ c = 4a × 3a

⇒ c = 12a²

Now, b² = 49a² and c =12a²

Hence, b² + c = 49a² + 12a²

⇒ b² + c = 61a²

Final answer must be a multiple of 61 & the multiple should be a perfect square.

Going through the options,

3721 = 61 × 61 (multiple is not a perfect square)

361 (not a multiple of 61)

427 = 61 × 7 (multiple is not a perfect square)

549 = 61 × 9 = 61 × 3²

Thus, possible value of b² + c = 549.

Hence, option (b).