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).