If the roots of the quadratic equation `x^2+p x+q=0` are `tan30^0` and `tan15^0,` respectively, then find the value of `q-p` .

Options

A. 2
B. 3
C. 0
D. 1

Explanation

To solve the problem, we need to find the value of \( q - p \) given that the roots of the quadratic equation \( x^2 + px + q = 0 \) are \( \tan 30^\circ \) and \( \tan 15^\circ \).

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots of the quadratic equation are given as:
\[ r_1 = \tan 30^\circ \quad \text{and} \quad r_2 = \tan 15^\circ \]

2. **Calculate the Values of the Roots**:
Using trigonometric values:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \quad \text{and} \quad \tan 15^\circ = 2 - \sqrt{3} \]

3. **Sum of the Roots**:
The sum of the roots \( r_1 + r_2 \) is given by:
\[ r_1 + r_2 = \tan 30^\circ + \tan 15^\circ = \frac{1}{\sqrt{3}} + (2 - \sqrt{3}) \]
Simplifying this:
\[ r_1 + r_2 = \frac{1}{\sqrt{3}} + 2 - \sqrt{3} \]

4. **Product of the Roots**:
The product of the roots \( r_1 \cdot r_2 \) is given by:
\[ r_1 \cdot r_2 = \tan 30^\circ \cdot \tan 15^\circ = \frac{1}{\sqrt{3}} \cdot (2 - \sqrt{3}) \]

5. **Using Vieta's Formulas**:
According to Vieta's formulas for the quadratic equation \( x^2 + px + q = 0 \):
- The sum of the roots is equal to \( -p \):
\[ -p = r_1 + r_2 \]
- The product of the roots is equal to \( q \):
\[ q = r_1 \cdot r_2 \]

6. **Set Up the Equations**:
From the above, we have:
\[ p = -\left(\frac{1}{\sqrt{3}} + 2 - \sqrt{3}\right) \]
\[ q = \frac{1}{\sqrt{3}}(2 - \sqrt{3}) \]

7. **Calculate \( q - p \)**:
Now we need to find \( q - p \):
\[ q - p = q + \left(\frac{1}{\sqrt{3}} + 2 - \sqrt{3}\right) \]
Substitute the value of \( q \):
\[ q - p = \frac{1}{\sqrt{3}}(2 - \sqrt{3}) + \left(\frac{1}{\sqrt{3}} + 2 - \sqrt{3}\right) \]

8. **Simplifying \( q - p \)**:
Combine the terms:
\[ q - p = \frac{1}{\sqrt{3}}(2 - \sqrt{3}) + \frac{1}{\sqrt{3}} + 2 - \sqrt{3} \]
\[ = \frac{2 - \sqrt{3} + 1}{\sqrt{3}} + 2 - \sqrt{3} \]
\[ = \frac{3 - \sqrt{3}}{\sqrt{3}} + 2 - \sqrt{3} \]

9. **Final Calculation**:
After simplification, we find:
\[ q - p = 1 \]

### Conclusion:
Thus, the value of \( q - p \) is \( 1 \).