If the randomly chosen point /dies between the roots of the equation `ax^2+bx+c=0`, then which of the following is correct ?

Options

A. `af(k)gt0`
B. `af(k)ge0`
C. `af(k)lt0`
D. all of these

Answer

To solve the problem, we need to analyze the given quadratic equation \( ax^2 + bx + c = 0 \) and its roots, which we will denote as \( \alpha \) and \( \beta \). The question states that a randomly chosen point lies between these roots. Let's go through the steps to understand what this implies.

### Step 1: Identify the Roots
The roots of the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \( D = b^2 - 4ac \) is the discriminant.

### Step 2: Conditions for Real Roots
For the roots \( \alpha \) and \( \beta \) to be real and distinct, the discriminant \( D \) must be greater than zero:
\[ D > 0 \implies b^2 - 4ac > 0 \]
If \( D = 0 \), the roots are real and equal, and if \( D < 0 \), the roots are complex.

### Step 3: Determine the Position of the Random Point
If a randomly chosen point lies between the roots \( \alpha \) and \( \beta \), it means that:
\[ \alpha < x < \beta \]
This implies that the value of the quadratic function at this point must be less than zero:
\[ f(x) = ax^2 + bx + c < 0 \]
for \( x \) in the interval \( (\alpha, \beta) \).

### Step 4: Analyze the Sign of the Quadratic Function
The quadratic function \( f(x) \) opens upwards if \( a > 0 \) and downwards if \( a < 0 \). Therefore:
- If \( a > 0 \), the function will be negative between the roots and positive outside.
- If \( a < 0 \), the function will be positive between the roots and negative outside.

### Conclusion
Thus, if a randomly chosen point lies between the roots of the equation \( ax^2 + bx + c = 0 \), the function value at that point is negative when \( a > 0 \) and positive when \( a < 0 \).

### Final Answer
The correct statement is:
- If \( a > 0 \), then \( f(x) < 0 \) for \( x \) between the roots.
- If \( a < 0 \), then \( f(x) > 0 \) for \( x \) between the roots.