Every quadratic equation `ax^(2) + bx + c = 0. where a. `b c in R. a != 0` has

Options

A. Exactly one real roots
B. At least one real root
C. At lest two real roots
D. At most two real roots

Answer

To determine the nature of the roots of the quadratic equation \( ax^2 + bx + c = 0 \) where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \), we will analyze the discriminant of the quadratic equation.

### Step-by-Step Solution:

1. **Identify the Quadratic Equation**:
The given quadratic equation is \( ax^2 + bx + c = 0 \).

2. **Calculate the Discriminant**:
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]

3. **Analyze the Discriminant**:
- If \( D > 0 \): The roots are **real and distinct**. This means there are two different real roots.
- If \( D = 0 \): The roots are **real and equal**. This means there is one real root that is repeated (also known as a double root).
- If \( D < 0 \): The roots are **imaginary and unequal**. This means there are no real roots, and the roots are complex conjugates.

4. **Conclusion**:
Therefore, for any quadratic equation \( ax^2 + bx + c = 0 \) where \( a, b, c \) are real numbers and \( a \neq 0 \):
- The maximum number of real roots is **2**.
- The roots can be real and distinct, real and equal, or imaginary and unequal depending on the value of the discriminant.

### Final Answer:
The maximum number of real roots of the quadratic equation \( ax^2 + bx + c = 0 \) is **2**.

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