In solving a prolem that reduces to a quadratic equation. One student makes a mistake in the constant term and obtain 8 and 2 for rotos. Another student makes a mistake only in the dcorfficient of first degree term and finds - 9 and - 1 for roots.The correct equation is

Options

A. (A)`x^(2) - 10x + 9 = 0`
B. (B)`x^(2) + 10x + 9 = 0`
C. (C)`x^(2) - 10x + 6 = 0`
D. (D)`x^(2) - 8x - 9 = 0`

Answer

To solve the problem, we need to find the correct quadratic equation based on the information provided about the roots obtained by two students.

### Step-by-Step Solution:

1. **Identify the roots from the first student**:
- The first student found the roots to be 8 and 2.
- The sum of the roots (S1) can be calculated as:
\[ S1 = 8 + 2 = 10 \]
- The product of the roots (P1) is:
\[ P1 = 8 \times 2 = 16 \]
- However, since the first student made a mistake in the constant term, we will only use the sum of the roots.

2. **Identify the roots from the second student**:
- The second student found the roots to be -9 and -1.
- The sum of the roots (S2) is:
\[ S2 = -9 + (-1) = -10 \]
- The product of the roots (P2) is:
\[ P2 = -9 \times -1 = 9 \]
- The second student made a mistake in the coefficient of the first-degree term, so we will only use the product of the roots.

3. **Formulate the correct quadratic equation**:
- We know the correct sum of the roots is 10 (from the first student) and the correct product of the roots is 9 (from the second student).
- The standard form of a quadratic equation is given by:
\[ f(x) = x^2 - (sum \ of \ roots) \cdot x + (product \ of \ roots) \]
- Substituting the values we found:
\[ f(x) = x^2 - 10x + 9 \]

4. **Set the equation to zero**:
- Therefore, the correct quadratic equation is:
\[ x^2 - 10x + 9 = 0 \]

### Final Answer:
The correct quadratic equation is:
\[ x^2 - 10x + 9 = 0 \]