Question

If the roots of \( x^2 - ax + b = 0 \) are two consecutive odd integers, then \( a^2 - 4b \) is:

• A. \( 3 \)
• B. \( 4 \)
• C. \( 5 \)
• D. \( 6 \)
• E. \( 7 \)

Hint:

Discriminant measures the "square of the distance" between roots. If roots are separated by $k$, the discriminant is $k^2$.

Answer 1

The Correct Option is B

Concept: For a quadratic equation \( x^2 - Sx + P = 0 \), the discriminant \( \mathcal{D} = S^2 - 4P \) is related to the difference between the roots \( (\alpha - \beta) \) by the identity: \[ \mathcal{D} = (\alpha - \beta)^2 \]

Step 1: Identify the roots.
Let the two consecutive odd integers be \( \alpha = 2n + 1 \) and \( \beta = 2n - 1 \) (or any \( k \) and \( k+2 \)).

Step 2: Calculate the difference between the roots.
\[ \alpha - \beta = (2n + 1) - (2n - 1) = 2 \] The distance between any two consecutive odd (or even) integers is always 2.

Step 3: Relate to the discriminant.
Given the equation \( x^2 - ax + b = 0 \), the discriminant is \( a^2 - 4b \). \[ a^2 - 4b = (\alpha - \beta)^2 \] \[ a^2 - 4b = (2)^2 = 4 \]