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 relationships between the roots (\(\alpha\), \(\beta\)) and the coefficients (a, b, c) of the quadratic equation are:

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

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

• \(\alpha + \beta < 0\), which means \(b > 0\) because \(a > 0\).
• \(\alpha \beta < 0\), therefore \(c < 0\) as \(a > 0\).

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

• Since \(\alpha \beta = \frac{c}{a} < 0\) and \(\alpha + \beta = -\frac{b}{a} < 0\), the statement \(a + b + c > 0\) is not always true.

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

• Testing by substituting values for \(\alpha\) and \(\beta\):
• \(a - b + c < 0\) because \(c < 0\).

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