Question

The equation \( x^2 + ax + 1 = 0 \) has no real roots. Which one of the following is correct?

• A. \( a \leq 2 \)
• B. \( a \geq 2 \)
• C. \( -2 \leq a<2 \)
• D. \( -2<a<2 \)

Hint:

For quadratic equations to have no real roots, the discriminant must be negative. Always check for the condition \( \Delta<0 \) to identify such cases.

Answer 1

The Correct Option is D

The given quadratic equation is: \[ x^2 + ax + 1 = 0 \] To determine the condition for no real roots, we use the discriminant \( \Delta \) of the quadratic equation, which is given by: \[ \Delta = b^2 - 4ac \] For the equation \( x^2 + ax + 1 = 0 \), we have:
\( a = 1 \),
\( b = a \),
\( c = 1 \).
So, the discriminant is: \[ \Delta = a^2 - 4 \cdot 1 \cdot 1 = a^2 - 4 \] For the equation to have no real roots, the discriminant must be negative, i.e.: \[ a^2 - 4<0 \] Solving for \( a \): \[ a^2<4 \] \[ -2<a<2 \] Thus, the correct condition for no real roots is \( -2<a<2 \).