Question

If the quadratic equation \( ax^2 + bx + c = 0 \) (\( a > 0 \)) has two roots \( \alpha \) and \( \beta \) such that \( \alpha < -2 \) and \( \beta > 2 \), then:

• A. \( c < 0 \)
• B. \( a + b + c > 0 \)
• C. \( a - b + c < 0 \)
• D. \( a - b + c > 0 \)

Answer 1

The Correct Option is A, C

1. The sum and product of the roots of the quadratic equation are:

\[ \alpha + \beta = -\frac{b}{a}, \quad \alpha \beta = \frac{c}{a}. \]

2. Given \(\alpha < -2\) and \(\beta > 2\):

• \(\alpha + \beta < 0\), implying \(b > 0\) since \(a > 0\).
• \(\alpha \beta < 0\), implying \(c < 0\) because \(a > 0\).

3. Consider \(a + b + c\):

• Since \(\alpha \beta = \frac{c}{a} < 0\) and \(\alpha + \beta = -\frac{b}{a} < 0\), \(a + b + c > 0\) does not hold in general.

4. Consider \(a - b + c\):

• Substitute the values of \(\alpha\) and \(\beta\) to test:
• \(a - b + c < 0\), as \(c < 0\).

Thus, the correct answers are (A) and (C).