If the minimum value of the quadratic function is 0, then which of the following is not definitely true, about the roots?

Options

A. Real
B. Non-Real'
C. Equal
D. Rational

Answer

To solve the problem, we need to analyze the properties of a quadratic function given that its minimum value is 0. A quadratic function can generally be expressed in the form:

\[ f(x) = ax^2 + bx + c \]

where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

### Step 1: Understanding the Minimum Value
The minimum value of a quadratic function occurs at the vertex of the parabola. The x-coordinate of the vertex can be calculated using the formula:

\[ x = -\frac{b}{2a} \]

Given that the minimum value of the function is 0, we can conclude that:

\[ f\left(-\frac{b}{2a}\right) = 0 \]

This means that the function touches the x-axis at the vertex.

### Step 2: Analyzing the Roots
Since the minimum value is 0, the quadratic function has a vertex that lies on the x-axis. The conditions for the roots of the quadratic function can be analyzed using the discriminant \( D \):

\[ D = b^2 - 4ac \]

1. If \( D > 0 \): There are two distinct real roots.
2. If \( D = 0 \): There is one real root (a repeated root).
3. If \( D < 0 \): There are no real roots (the roots are complex).

### Step 3: Conclusion About the Roots
Since the minimum value is 0, we know that the quadratic function touches the x-axis at the vertex. This implies that:

- The quadratic has at least one real root (the vertex).
- The roots are equal (since it only touches the x-axis).

### Step 4: Evaluating the Options
Now, let's evaluate the options given in the question:

1. **Real Roots**: This is definitely true because the function touches the x-axis.
2. **Non-Real Roots**: This is not true since we established that there is at least one real root.
3. **Equal Roots**: This is definitely true because the minimum value is 0, indicating a double root.
4. **Rational Roots**: This is not necessarily true. The roots can be irrational even if they are equal.

### Final Answer
The option that is **not definitely true** about the roots is:

**Non-real roots** (Option 2) and **Rational roots** (Option 4).