In solving quadratic equation `x^(2) + px + q = 0 `, one student makes mistake only in the constant term obtains 4 and 3 as the roots. Another students makes a mistake only in the coefficient of x and finds - 5 and - 2 as the roots. Determine the correct equation

Answer

To solve the problem, we need to determine the correct quadratic equation based on the information given about the mistakes made by two students.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have a quadratic equation of the form \( x^2 + px + q = 0 \).
- The first student finds the roots to be 4 and 3, but makes a mistake in the constant term \( q \).
- The second student finds the roots to be -5 and -2, but makes a mistake in the coefficient of \( x \) (which is \( p \)).

2. **Using the First Student's Roots**:
- The roots are 4 and 3.
- The sum of the roots \( S_1 = 4 + 3 = 7 \).
- The product of the roots \( P_1 = 4 \times 3 = 12 \).
- According to Vieta's formulas, for the equation \( x^2 + px + q = 0 \):
- The sum of the roots \( S = -p \) implies \( p = -7 \).
- The product of the roots \( P = q \) implies \( q = 12 \).

3. **Using the Second Student's Roots**:
- The roots are -5 and -2.
- The sum of the roots \( S_2 = -5 + (-2) = -7 \).
- The product of the roots \( P_2 = (-5) \times (-2) = 10 \).
- According to Vieta's formulas:
- The sum of the roots \( S = -p \) implies \( p = 7 \).
- The product of the roots \( P = q \) implies \( q = 10 \).

4. **Identifying the Mistakes**:
- The first student correctly calculated \( p \) but made a mistake in \( q \). So, \( p = -7 \) is correct.
- The second student correctly calculated \( q \) but made a mistake in \( p \). So, \( q = 10 \) is correct.

5. **Forming the Correct Equation**:
- The correct values for \( p \) and \( q \) are:
- \( p = -7 \)
- \( q = 10 \)
- Therefore, the correct quadratic equation is:
\[ x^2 - 7x + 10 = 0 \]

### Final Answer:
The correct quadratic equation is:
\[ x^2 - 7x + 10 = 0 \]