if p,q,r are real numbers satisfying the condition p + q +r =0 , then the roots of the quadratic equation `3px^(2) + 5qx + 7r =0` are

Options

A. positive
B. negative
C. Real and distinct
D. Imaginary

Answer

To solve the problem, we need to find the roots of the quadratic equation \(3px^2 + 5qx + 7r = 0\) given the condition \(p + q + r = 0\).

### Step-by-step Solution:

1. **Given Condition**:
We start with the condition provided:
\[ p + q + r = 0 \]
From this, we can express \(q\) in terms of \(p\) and \(r\):
\[ q = -p - r \]

2. **Substituting \(q\)**:
Now, we substitute \(q\) into the quadratic equation:
\[ 3px^2 + 5(-p - r)x + 7r = 0 \]
Simplifying this gives:
\[ 3px^2 - 5(p + r)x + 7r = 0 \]
or
\[ 3px^2 - 5px - 5rx + 7r = 0 \]

3. **Identifying Coefficients**:
From the equation \(3px^2 - 5(p + r)x + 7r = 0\), we identify:
- \(A = 3p\)
- \(B = -5(p + r)\)
- \(C = 7r\)

4. **Finding the Discriminant**:
The discriminant \(D\) of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:
\[ D = B^2 - 4AC \]
Substituting our coefficients:
\[ D = (-5(p + r))^2 - 4(3p)(7r) \]
Simplifying this:
\[ D = 25(p + r)^2 - 84pr \]

5. **Expanding \((p + r)^2\)**:
Expanding \((p + r)^2\):
\[ D = 25(p^2 + 2pr + r^2) - 84pr \]
This simplifies to:
\[ D = 25p^2 + 50pr + 25r^2 - 84pr \]
Combining like terms:
\[ D = 25p^2 + 25r^2 - 34pr \]

6. **Analyzing the Discriminant**:
We need to analyze the discriminant \(D\):
\[ D = 25(p^2 + r^2) - 34pr \]
We can rewrite this as:
\[ D = 25\left(p^2 + r^2 - \frac{34}{25}pr\right) \]
The expression \(p^2 + r^2 - \frac{34}{25}pr\) can be analyzed further.

7. **Using AM-GM Inequality**:
By applying the AM-GM inequality:
\[ p^2 + r^2 \geq 2pr \]
Thus:
\[ p^2 + r^2 - \frac{34}{25}pr \geq 2pr - \frac{34}{25}pr = \left(2 - \frac{34}{25}\right)pr = \left(\frac{50 - 34}{25}\right)pr = \frac{16}{25}pr \]
Since \(pr\) can take any real value, we conclude that \(D\) can be non-negative.

8. **Conclusion on Roots**:
Since the discriminant \(D\) is non-negative, the roots of the quadratic equation are real. Depending on the specific values of \(p\) and \(r\), they can be either real and distinct or real and equal.

### Final Answer:
The roots of the quadratic equation \(3px^2 + 5qx + 7r = 0\) are **real**.