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

Options

A. exactly one real root.
B. at least one rea root.
C. at least two real roots.
D. at most two real roots.

Explanation

To solve the question regarding the nature of the roots of the quadratic equation \( ax^2 + bx + c = 0 \), we need to analyze the discriminant of the quadratic equation. The discriminant \( D \) is given by the formula:

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

### Step 1: Identify the discriminant
The discriminant helps us determine the nature of the roots of the quadratic equation. We will evaluate the discriminant for different cases.

### Step 2: Analyze the discriminant
1. **If \( D > 0 \)**: The equation has **two distinct real roots**.
2. **If \( D = 0 \)**: The equation has **exactly one real root** (also called a repeated root).
3. **If \( D < 0 \)**: The equation has **no real roots** (the roots are complex).

### Step 3: Conclusion
Since the question asks for the nature of roots for every quadratic equation \( ax^2 + bx + c = 0 \) where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \):
- A quadratic equation can have **at most two real roots** (when \( D \geq 0 \)).
- It can have **at least one real root** (when \( D \geq 0 \)).
- It can have **exactly one real root** (when \( D = 0 \)).
- It can have **no real roots** (when \( D < 0 \)).

Thus, the correct answer to the question is: **at most two real roots**.

### Final Answer:
Every quadratic equation \( ax^2 + bx + c = 0 \) where \( a, b, c \in \mathbb{R} \) has at most two real roots.

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