Question:
For what values of $k$, the equation $k x^{2}-6 x-2=0$ has real roots?
(a) $k \leq \frac{-9}{2}$
(b) $k \geq \frac{-9}{2}$
(c) $k \leq-2$
(d) None of these
Solution:
(b) $k \geq \frac{-9}{2}$
It is given that the roots of the equation $\left(k x^{2}-6 x-2=0\right)$ are real.
$\therefore D \geq 0$
$\Rightarrow\left(b^{2}-4 a c\right) \geq 0$
$\Rightarrow(-6)^{2}-4 \times k \times(-2) \geq 0$
$\Rightarrow 36+8 k \geq 0$
$\Rightarrow k \geq \frac{-36}{8}$
$\Rightarrow k \geq \frac{-9}{2}$
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