Two candidates attempt to solve a quadratic equation of the form `x^(2)+px+q=0`. One starts with a wrong value of p and finds the roots to be 2 and 6. the other starts with a wrong vlaue of q and finds the roots to be 2 and -9. find the correct roots and the equation.

Answer

To solve the problem, we need to find the correct roots and the quadratic equation based on the information given about the roots found by the two candidates.

### Step-by-Step Solution:

1. **Identify the roots from the first candidate**:
The first candidate finds the roots to be 2 and 6.
- The sum of the roots \( S_1 = 2 + 6 = 8 \).
- The product of the roots \( P_1 = 2 \times 6 = 12 \).

2. **Form the equation with the first candidate's roots**:
The quadratic equation can be formed using the roots:
\[ x^2 - S_1 x + P_1 = 0 \implies x^2 - 8x + 12 = 0 \]

3. **Identify the roots from the second candidate**:
The second candidate finds the roots to be 2 and -9.
- The sum of the roots \( S_2 = 2 + (-9) = -7 \).
- The product of the roots \( P_2 = 2 \times (-9) = -18 \).

4. **Form the equation with the second candidate's roots**:
The quadratic equation can be formed using the roots:
\[ x^2 - S_2 x + P_2 = 0 \implies x^2 + 7x - 18 = 0 \]

5. **Set up the equations based on the candidates' findings**:
- From the first candidate, we have \( p \) is incorrect, but \( q \) is correct. The correct \( q \) should be 12.
- From the second candidate, \( q \) is incorrect, but \( p \) is correct. The correct \( p \) should be 7.

6. **Combine the information**:
We have established that the correct quadratic equation is:
\[ x^2 + 7x + 12 = 0 \]

7. **Find the correct roots**:
To find the correct roots, we can factor the equation:
\[ x^2 + 7x + 12 = (x + 3)(x + 4) = 0 \]
Thus, the roots are:
\[ x = -3 \quad \text{and} \quad x = -4 \]

### Final Answer:
The correct roots of the quadratic equation are \( -3 \) and \( -4 \), and the correct equation is:
\[ x^2 + 7x + 12 = 0 \]