Question

The coefficients $a,b$ and $c$ of the quadratic equation, $a{x}^2+bx+c=0$ are 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 $\frac{5}{216}$

$a{x}^2+bx+c=0$
$a,b,c\in \{1,2,3,4,5,6\}$
$n\left(S\right)=6\times 6\times 6=216$
$D=0\Rightarrow b^2=4ac$
$\Rightarrow ac=\frac{b^2}{4}$
If $b=2,ac=1\Rightarrow a=1,c=1$
If $b=4,ac=4$
$\Rightarrow a=1,c=4$
or $a=4,c=1$
or $a=2,c=2$
If $b=6,ac=9\Rightarrow a=3,c=3$
$\therefore$ Probability $=\frac{5}{216}$