Let $S$ be the set of all quadratic equations of the form $x^2+b x+c=0$, where $b, c \in\{1,2,3$, $4,5,6\}$.…
Let $S$ be the set of all quadratic equations of the form $x^2+b x+c=0$, where $b, c \in\{1,2,3$, $4,5,6\}$. If an equation is selected at random from $S$, then the probability that the equation has real roots is
$\frac{9}{12}$
$\frac{9}{36}$
$\frac{19}{36}$
$\frac{7}{36}$
Solution
Given $S=\left\{x^2+b x+c=0\right.$;
$
b, c \in\{1,2,3,4,5,6\}\}
$
Since, we know that a quadratic equation $a x^2+b x+c=0$ has real roots, if $b^2-4 a c \geq 0$
$\because a=1$, hence condition for the given equation is
$
b^2-4 c \geq 0
$
Total number of equations in $S$, which has roots
$
\begin{aligned}
& =6 \times 6 \\
n(S) & =36
\end{aligned}
$
Let $E$ be the event, which contains equations having real roots following condition $b^2 \geq 4 C$
$
\begin{array}{ll}
\therefore & n(E)=19 \\
\therefore & P(E)=\frac{n(E)}{n(S)}=\frac{19}{36}
\end{array}
$