Question

If $a>0$ and the discriminant of $a{x}^2+2bx+c$ is negative, then

$\left|\begin{array}{ccc}a& b& ax+b\\ b& c& bx+c\\ ax+b& bx+c& 0\end{array}\right|$ is

• A. positive
• B. equal to $(ac-b^2)(a{x}^2+2bx+c)$
• C. negative
• D. equal to $0$

Answer 1

The correct option is C negative

We have $4b^2-4ac<0\therefore b^2-ac<0$
$=\left|\begin{array}{ccc}a& b& ax+b\\ b& c& bx+c\\ 0& 0& -(a{x}^2+2bx+c)\end{array}\right|[R_3\to R_3-(x{R}_1+R_2\left)\right]=(a{x}^2+2bx+c)(b^2-ac)$
$=(+)(-)=$ negative