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).
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