Form the quadratic equation if its root are `1/2, -1/2`

Answer

To form a quadratic equation given its roots, we can use the relationship between the roots and the coefficients of the quadratic equation. Let's denote the roots as \( \alpha \) and \( \beta \). In this case, the roots are given as \( \alpha = \frac{1}{2} \) and \( \beta = -\frac{1}{2} \).

### Step-by-Step Solution:

1. **Identify the roots**:
We have the roots \( \alpha = \frac{1}{2} \) and \( \beta = -\frac{1}{2} \).

2. **Calculate the sum of the roots**:
The sum of the roots \( \alpha + \beta \) is calculated as follows:
\[ \alpha + \beta = \frac{1}{2} + \left(-\frac{1}{2}\right) = 0 \]

3. **Calculate the product of the roots**:
The product of the roots \( \alpha \beta \) is calculated as follows:
\[ \alpha \beta = \frac{1}{2} \times \left(-\frac{1}{2}\right) = -\frac{1}{4} \]

4. **Form the quadratic equation**:
The general form of a quadratic equation with roots \( \alpha \) and \( \beta \) is given by:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values we calculated:
\[ x^2 - (0)x - \frac{1}{4} = 0 \]
This simplifies to:
\[ x^2 - \frac{1}{4} = 0 \]

5. **Eliminate the fraction**:
To eliminate the fraction, we can multiply the entire equation by 4:
\[ 4x^2 - 1 = 0 \]

### Final Result:
The quadratic equation formed from the roots \( \frac{1}{2} \) and \( -\frac{1}{2} \) is:
\[ 4x^2 - 1 = 0 \]