Question

If $\frac{7}{2}$ and $1$ are the roots of the equation

$\left|\begin{array}{ccc}2x& 3& 7\\ 2& 2x& 2\\ 7& 6& 2x\end{array}\right|=0$, then the third root is

• A. $\frac{-5}{2}$
• B. $\frac{-7}{2}$
• C. $\frac{-9}{2}$
• D. $\frac{-3}{2}$

Answer 1

The correct option is C $\frac{-9}{2}$

$\left|\begin{array}{ccc}2x& 3& 7\\ 2& 2x& 2\\ 7& 6& 2x\end{array}\right|=0$

Use $R_3\to R_3+R_2+R_1$
$\left|\begin{array}{ccc}2x& 3& 7\\ 2& 2x& 2\\ 9+2x& 9+2x& 9+2x\end{array}\right|=0$
So, if $x=-\frac{9}{2}$, then the determinant will be
$\left|\begin{array}{ccc}-9& 3& 7\\ 2& -9& 2\\ 0& 0& 0\end{array}\right|=0$