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. $\frac{5}{216}$
• B. $\frac{3}{216}$
• C. $\frac{15}{216}$
• D. $\frac{1}{216}$

Answer 1

Answer: (A)

$\frac{5}{216}$

Roots equal $\Rightarrow b^2-4ac=0$
${\left(\frac{b}{2}\right)}^2=ac$
Each coefficient is an integer, so we consider the following cases:
If $b=1$
$\therefore \frac{1}{4}=ac.$ No integral values of a and c.

If $b=2$
$1=ac$$\therefore (1,1)$

If $b=3$
$\frac{9}{2}=ac$ No integral values of a and c.

If $b=4$
$4=ac\therefore (1,4),(2,2),(4,1),$

If $b=5$
$\frac{25}{2}=ac,$ No integral values of a and c.

$b=6$
$9=ac\therefore (3,3)$
Thus we have 5 favourable ways for $b=2,4,6$
Total number of equations is $6\times 6\times 6=216$ as each variable has 6 possible results, when a die is thrown.
$\therefore$ Required probability is $\frac{5}{216}$