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  4. The coefficients a , b and c of the quadratic equation, a x2+b x+c=0 are obtained by throwing a dice three times. The probability that this equation has equal roots is

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Q. 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

JEE MainJEE Main 2021Complex Numbers and Quadratic Equations

Solution:

$a x^{2}+b x+c=0$
For equal roots $D =0$
$\Rightarrow b ^{2}=4 ac$
$Case I : ac =1$
$( a , b , c )=(1,2,1)$
$Case II : ac = 4 $
$(a, b, c)=(1,4,4)$
or $(4,4,1)$
or $ (2,4,2)$
$Case III : ac =9$
$(a, b, c)=(3,6,3)$
Required probability $=\frac{5}{216}$