# The coefficients a, b and c

Question:

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

1. (1) $\frac{1}{54}$

2. (2) $\frac{1}{72}$

3. (3) $\frac{1}{36}$

4. (4) $\frac{5}{216}$

Correct Option: , 4

Solution:

$a x^{2}+b x+c=0$

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

$\mathrm{n}(\mathrm{s})=6 \times 6 \times 6=216$

$\mathrm{D}=0 \Rightarrow \mathrm{b}^{2}=4 \mathrm{ac}$

$\mathrm{ac}=\frac{\mathrm{b}^{2}}{4} \quad$ If $\mathrm{b}=2, \mathrm{ac}=1 \quad \Rightarrow \quad \mathrm{a}=1, \mathrm{c}=1$

If $b=4, a c=4 \quad \Rightarrow \quad a=1, c=4$

$\mathrm{a}=4, \mathrm{c}=1$

Ifb $=6, \mathrm{ac}=9 \Rightarrow \quad 2, \mathrm{c}=2$

$\therefore$ probability $=\frac{5}{216}$