Question

If x = –9 is a root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0,$ then other two roots are ___________.

Answer 1

Given: x = –9 is a root of $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|=0$

Let ∆ = $\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right|$



$$\begin{gathered} ∆=\left|\begin{array}{ccc}x& 3& 7\\ 2& x& 2\\ 7& 6& x\end{array}\right| \\ \mathrm{Applying}{R}_1\to R_1+R_2+R_3 \\ =\left|\begin{array}{ccc}x+2+7& 3+x+6& 7+2+x\\ 2& x& 2\\ 7& 6& x\end{array}\right| \\ =\left|\begin{array}{ccc}x+9& x+9& x+9\\ 2& x& 2\\ 7& 6& x\end{array}\right| \\ \mathrm{Taking}\mathrm{out}\left(x+9\right)\mathrm{common}\mathrm{from}{R}_1 \\ =\left(x+9\right)\left|\begin{array}{ccc}1& 1& 1\\ 2& x& 2\\ 7& 6& x\end{array}\right| \\ \mathrm{Applying}{C}_2\to C_2-C_1\mathrm{and}{C}_3\to C_3-C_1 \\ =\left(x+9\right)\left|\begin{array}{ccc}1& 1-1& 1-1\\ 2& x-2& 2-2\\ 7& 6-7& x-7\end{array}\right| \\ =\left(x+9\right)\left|\begin{array}{ccc}1& 0& 0\\ 2& x-2& 0\\ 7& -1& x-7\end{array}\right| \\ \mathrm{Expanding}\mathrm{through}{R}_1 \\ =\left(x+9\right)\left(1\left(x-2\right)\left(x-7\right)\right) \\ =\left(x+9\right)\left(x-2\right)\left(x-7\right) \\ \mathrm{Since},∆=0 \\ \Rightarrow \left(x+9\right)\left(x-2\right)\left(x-7\right)=0 \\ \Rightarrow x=-9,2,7 \end{gathered}$$

Hence, other two roots are 2 and 7.