Question

If both a and b belong to the set $\{1, 2, 3, 4\}$, then the number of equations of the form $ax^2 + bx + 1 = 0$ having real roots is:

• A. 10
• B. 7
• C. 6
• D. 12

Hint:

For quadratic equations, check the discriminant to determine if real roots exist.

Answer 1

The Correct Option is C

The discriminant of the quadratic equation $ax^2 + bx + 1 = 0$ is: \[ \Delta = b^2 - 4ac \] For real roots, the discriminant must be non-negative, i.e., $\Delta \geq 0$. Here $a$ and $b$ belong to $\{1, 2, 3, 4\}$, so we check the discriminant for each pair of $a$ and $b$. The discriminant for each combination of $a$ and $b$ is calculated, and we find that there are 6 combinations where the discriminant is non-negative. Thus, the number of equations with real roots is 6.