Practicing Success
If $a > 0$ and discriminant of $ax^2 + 2bx + c$ is negative, then $Δ=\begin{vmatrix}a&b&ax+b\\b&c&bx+c\\ax+b&bx+c&0\end{vmatrix}$, is |
positive $(ac-b^2) (ax^2 + 2bx + c)$ negative 0 |
negative |
It is given that the discriminant of $ax^2 + 2bx + c$ is negative. $∴4b^2-4ac <0⇒b^2-ac <0⇒ ac-b^2>0$ ...(i) Also, a > 0 and discriminant is negative. $∴ax^2+2bx + c > 0$ for all $x ∈ R$ ...(ii) Now, $Δ=\begin{vmatrix}a&b&ax+b\\b&c&bx+c\\ax+b&bx+c&0\end{vmatrix}$ $⇒Δ=\begin{vmatrix}a&b&0\\b&c&0\\ax+b&bx+c&-x(ax+b)-(bx+c)\end{vmatrix}$ $⇒Δ=\begin{vmatrix}a&b&0\\b&c&0\\ax+b&bx+c&-(ax^2+2bx+c)\end{vmatrix}$ $⇒Δ=-(ax^2+2bx + c) (ac-b^2) <0$ [Using (i) and (ii)] |