If a quadratic equation has one real root then the other root will necessarily be

Options

A. real
B. equal
C. complex conjugate
D. rational and unequal

Answer

To determine the nature of the other root of a quadratic equation when one root is known to be real, we can follow these steps:

### Step 1: Understand the nature of roots in a quadratic equation
A quadratic equation is generally expressed in the form \( ax^2 + bx + c = 0 \). The roots of this equation can be determined using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{D}}{2a} \]
where \( D \) is the discriminant given by \( D = b^2 - 4ac \).

### Step 2: Analyze the discriminant
The discriminant \( D \) helps us understand the nature of the roots:
- If \( D > 0 \): The equation has two distinct real roots.
- If \( D = 0 \): The equation has exactly one real root (a repeated root).
- If \( D < 0 \): The equation has two complex conjugate roots.

### Step 3: Given one real root
If we know that one of the roots (let's call it \( \alpha \)) is real, we can analyze the implications:
- If \( \alpha \) is real, then we can consider the discriminant \( D \).
- If \( D \geq 0 \) (which includes \( D = 0 \)), the other root (let's call it \( \beta \)) must also be real. This is because having one real root implies that the quadratic does not "turn" into the complex plane.

### Step 4: Conclusion
Thus, if a quadratic equation has one real root, the other root must necessarily also be real. Therefore, the answer is that the other root will be real.

### Final Answer
If a quadratic equation has one real root, then the other root will necessarily be **real**.

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