Question

The coefficients $a,b\text{and}c$ of the quadratic equation $a{x}^2+bx+c=0$ is obtained by throwing a dice three times. The probability that this equation has equal roots is

• A. $\frac{1}{54}$
• B. $\frac{1}{72}$
• C. $\frac{1}{36}$
• D. $\frac{5}{216}$

Answer 1

The correct option is D

$\frac{5}{216}$

Explanation for the Correct Answer:

Finding the probability that this equation has equal roots:

Given quadratic equation is $a{x}^2+bx+c=0$

Now, the possible values of $a,b\text{and}c$ are possible values on dice which is

$a,b,c\in \left\{1,2,3,4,5,6\right\}$

Total Number of Outcomes in Throwing Dice Three times is $$\begin{gathered} N_{\text{Total outcome}}=6\times 6\times 6 \\ =216 \end{gathered}$$

For equal roots, We know the condition such that $b^2=4ac$.

When $b=4$, there are three possible ways

$$\begin{gathered} a=1\text{and}c=4 \\ a=4\text{and}c=1 \\ a=2\text{and}c=2 \end{gathered}$$

and when $b=6$ ,there is one possible way

$\Rightarrow a=3\text{and}c=3$

and when $b=2$, there is one possible way

$\Rightarrow a=1\text{and}c=1$ which in total One Case

So, the total possible cases are $N_{\text{Possible outcome}}=5$

So, probability is

$$\begin{gathered} P=\frac{N_{\text{Possible Outcome}}}{N_{\text{Total Outcome}}} \\ P=\frac{5}{216} \end{gathered}$$

Hence, the correct answer is Option (D)