Question:
Consider the quadratic equation $x^{2}+bx+c=0$ where the coefficients $b,c\in\{2,6,8,9\}$. Find the probability that the equation has real and equal roots:}
Consider the quadratic equation $x^{2}+bx+c=0$ where the coefficients $b,c\in\{2,6,8,9\}$. Find the probability that the equation has real and equal roots:}
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Equal roots occur only when the discriminant is zero.
Updated On: Jun 12, 2026
- $\frac{1}{16}$
- $\frac{1}{8}$
- $\frac{1}{4}$
- $\frac{3}{16}$
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The Correct Option is A
Solution and Explanation
Concept:
For equal roots,
$$
b^2-4c=0
$$
or
$$
b^2=4c
$$
Step 1: Count total possible pairs. Both $b$ and $c$ can take $4$ values. $$ n(S)=4\times4=16 $$
Step 2: Find favourable pairs. Checking all values of $b$:
• $b=2 \Rightarrow c=1$ (not available)
• $b=6 \Rightarrow c=9$ (available)
• $b=8 \Rightarrow c=16$ (not available)
• $b=9 \Rightarrow c=20.25$ (not available) Only one pair satisfies the condition: $$ (b,c)=(6,9) $$ Thus, $$ n(E)=1 $$
Step 3: Probability. $$ P(E)=\frac{1}{16} $$ \[ \boxed{\frac{1}{16}} \]
Step 1: Count total possible pairs. Both $b$ and $c$ can take $4$ values. $$ n(S)=4\times4=16 $$
Step 2: Find favourable pairs. Checking all values of $b$:
• $b=2 \Rightarrow c=1$ (not available)
• $b=6 \Rightarrow c=9$ (available)
• $b=8 \Rightarrow c=16$ (not available)
• $b=9 \Rightarrow c=20.25$ (not available) Only one pair satisfies the condition: $$ (b,c)=(6,9) $$ Thus, $$ n(E)=1 $$
Step 3: Probability. $$ P(E)=\frac{1}{16} $$ \[ \boxed{\frac{1}{16}} \]
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