Question

If x = – 9 is a root of `|(x, 3, 7),(2, x, 2),(7, 6, x)|` = 0, then other two roots are ______.

Answer 1

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(x^{2 –} 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.