Question

Absolute value of sum of roots of the equation $\left|\begin{array}{ccc}x+2& 2x+3& 3x+4\\ 2x+3& 3x+4& 4x+5\\ 3x+5& 5x+8& 10x+17\end{array}\right|=0$ is ...........

Answer 1

Applying $R_3\to (R_3-R_2);R_3\to (R_2-R_1)$
$\left|\begin{array}{ccc}(x+2)& (2x+3)& (3x+4)\\ (x+1)& (x+1)& (x+1)\\ (x+2)& 2(x+2)& 6(x+2)\end{array}\right|=0$
Applying $C_3\to C_3-C_2;C_2\to C_2-C_1$
$\Rightarrow \left|\begin{array}{ccc}(x+2)& (x+1)& (x+1)\\ (x+1)& 0& 0\\ (x+2)& (x+2)& 6(x+2)\end{array}\right|=0\Rightarrow -(x+1)\{4(x+1)(x+2)-(x+2)(x+2)\}=0\Rightarrow -3(x+1)\left\{(x+1)(x+2)\right\}=0\Rightarrow x=-1,-1,-2$
Hence sum of roots is $-4$ its absolute value is $4$