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Question
If x = – 9 is a root of `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0, then other two roots are ______.
Solution
If x = – 9 is a root of `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0, then other two roots are x = –9, 2, 7.
Explanation:
We have, `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0
Expanding along R1
⇒ `x|(x, 2),(6, x)| -3|(2, 2),(7, x)| + |(2, x),(7, 6)|` = 0
⇒ x(x2 – 12) – 3(2x – 14) + 7(12 – 7x) = 0
⇒ x3 – 12x – 6x + 42 + 84 – 49x = 0
⇒ x3 – 67x + 126 = 0 .....(1)
The roots of the equation may be the factors of 126
i.e., 2 × 7 × 9
9 is given the root of the determinant put x = 2 in equation (1)
(2)3 – 67 × 2 + 126
⇒ 8 – 134 + 126 = 0
Hence, x = 2 is the other root.
Now, put x = 7 in equation (1)
(7)3 – 67(7) + 126
⇒ 343 – 469 + 126 = 0
Hence, x = 7 is also the other root of the determinant.
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