Question

If $7$ and $2$ are two roots of the following equation $\left[\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right]=0$, then its third root will be

• A. $-9$
• B. $14$
• C. $\frac{1}{2}$
• D. None of these

Answer 1

The correct option is A $-9$

$A=\left[\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right]$

$Det\left(A\right)=x(x^2-12)-3(2x-14)+7(12-7x)$

$\Rightarrow 0=x^3-12x-6x+42+84-49x$

$0=x^3-67x+126$

Since the sum of roots is the coefficient of $x^2$, which is zero here,
Let $a$ be the third root.
So, $a+2+7=0$
the third root would be $-9.$