Question

Each coefficient in the equation $a{x}^2+bx+c=0$ is determined by throwing an ordinary die. Find the probability that the equation will have equal roots.

• A. $5/216$
• B. $10/216$
• C. $10/36$
• D. $5/36$

Answer 1

Answer: (A)

$5/216$

Roots equal $\Rightarrow b^2-4ac=0\Rightarrow {\left(\frac{b}{2}\right)}^2=ac$
Each coefficients is an integer, So we consider the following cases
$b=1\Rightarrow \frac{1}{4}=ac$ No integral value of $a$ and $c$
$b=2\Rightarrow 1=ac\therefore (1,1)$
$b=3\Rightarrow \frac{9}{2}=ac$ No integral value of $a$ and $c$
$b=4\Rightarrow 4=ac\therefore (1,4),(2,2),(4,1)$
$b=5\Rightarrow \frac{25}{2}=ac$ No integral value of $a$ and $c$
$b=6\Rightarrow 9=ac\therefore (3,3)$
Thus we have $5$ favorable ways for $b=2,4,6$
Total number of equation is $\mathrm{6.6.6}=216$
Therefore required probability is $\frac{5}{216}$.