Question

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

• A. $2,7$
• B. $-2,7$
• C. $2,-7$
• D. $-2,-7$

Answer 1

Answer: (A)

$2,7$

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

Simplifying matrix we get,

$x(x^2-12)-3(2x-14)+7(12-7x)=x^3-12x-6x+42+84-49x=x^3-67x+126$

The equation can be simplified by, $x^3-67x+126(x+9)(x^2-9x+14)$

$(x^2-9x+14)=(x^2-7x-2x+14)=(x-2)(x-7),x=2,7$

Hence the roots are $2,7,-9$.