The sum of the roots of the equation `x^2+ px + q= 0` is equal to the sum of their squares, then

Options

A. `p^(2)-q^(2) = Q`
B. `p^(2) + q^(2) = 2q`
C. `p^(2) + p = 2q`
D. `q^(2) + q= 2p`

Answer

To solve the problem, we need to find the relationship between the coefficients \( p \) and \( q \) given that the sum of the roots of the quadratic equation \( x^2 + px + q = 0 \) is equal to the sum of their squares.

### Step-by-Step Solution:

1. **Identify the roots**: Let the roots of the equation \( x^2 + px + q = 0 \) be \( \alpha \) and \( \beta \).

2. **Sum of the roots**: According to Vieta's formulas, the sum of the roots \( \alpha + \beta \) is given by:
\[ \alpha + \beta = -\frac{p}{1} = -p \]

3. **Product of the roots**: The product of the roots \( \alpha \beta \) is given by:
\[ \alpha \beta = \frac{q}{1} = q \]

4. **Sum of the squares of the roots**: The sum of the squares of the roots can be expressed as:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \]
Substituting the values from steps 2 and 3, we get:
\[ \alpha^2 + \beta^2 = (-p)^2 - 2q = p^2 - 2q \]

5. **Setting up the equation**: According to the problem, the sum of the roots is equal to the sum of their squares:
\[ \alpha + \beta = \alpha^2 + \beta^2 \]
Substituting the expressions we derived:
\[ -p = p^2 - 2q \]

6. **Rearranging the equation**: To find the relationship between \( p \) and \( q \), we rearrange the equation:
\[ p^2 + p - 2q = 0 \]

7. **Final form**: This can be rewritten as:
\[ p^2 + p = 2q \]

Thus, the correct relationship is:
\[ p^2 + p = 2q \]

### Conclusion:
The answer is option 3: \( p^2 + p = 2q \).