The coefficients $a, b, c$ in the quadratic equation $a x^2+b x+c=0$ are chosen from the set $\{1,2,3,4,5,6,7,8\}$. The probability of this equation having repeated roots is :
Solution
<p>Given quadratic equation is</p>
<p>$a x^2+b x+c=0 \text { where } a, b, c \in\{1,2,3, \ldots, 8\}$</p>
<p>For repeated roots,</p>
<p>$$\begin{aligned}
& b^2-4 a c=0 \\
& \Rightarrow b^2=4 a c
\end{aligned}$$</p>
<p>$\Rightarrow a c$ must be a perfect square</p>
<p>$$(a, c) \in\{(1,1),(1,4),(2,2),(2,8),(3,3),(4,1),(4,4),(5,5),(6,6),(7,7),(8,2),(8,8)\}$$</p>
<p>Corresponding $b$ must lie in set $\{1,2,3, \ldots 8\}$</p>
<p>$$\begin{aligned}
& (a, b, c) \in\{(1,2,1),(1,2,4),(2,4,2),(2,8,8) \\
& (3,6,3),(4,4,1),(4,8,4),(8,8,2)\} \\
& \therefore \text { probability }=\frac{8}{8^3} \\
& =\frac{1}{64} \\
&
\end{aligned}$$</p>
About this question
Subject: Mathematics · Chapter: Probability · Topic: Classical and Axiomatic Probability
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