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

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:

cdquestions Admin · Accepted Answer

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) .

Answer

Answer

$ a + b + c > 0 $

Answer

$ a - b + c < 0 $

Answer

$ a - b + c > 0 $

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:

The Correct Option is A, C

Solution and Explanation

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