Two students while solving a quadratic equation in x, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term and coefficient of `x^(2)` correctly as -6 and 1 respectively. The correct roots are

Options

A. 3,-2
B. `-3,2`
C. `-6,-1`
D. `6,-1`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Roots from the First Student:**
- The first student obtained the roots 3 and 2.
- According to the properties of quadratic equations, the sum of the roots (α + β) is given by:
\[ \alpha + \beta = 3 + 2 = 5 \]

2. **Understanding the Roots from the Second Student:**
- The second student copied the constant term (c) and the coefficient of \(x^2\) (a) correctly as -6 and 1, respectively.
- Therefore, the quadratic equation can be represented as:
\[ x^2 + bx - 6 = 0 \]
- The product of the roots (α * β) is given by:
\[ \alpha \cdot \beta = -6 \]

3. **Setting Up the Equations:**
- From the first student's roots, we have:
\[ \alpha + \beta = 5 \quad \text{(1)} \]
- From the second student's roots, we have:
\[ \alpha \cdot \beta = -6 \quad \text{(2)} \]

4. **Using the Sum and Product of Roots:**
- We can use the equations (1) and (2) to form a quadratic equation. The general form of a quadratic equation based on sum and product of roots is:
\[ x^2 - (\alpha + \beta)x + (\alpha \cdot \beta) = 0 \]
- Substituting the values from equations (1) and (2):
\[ x^2 - 5x - 6 = 0 \]

5. **Factoring the Quadratic Equation:**
- To factor the equation \(x^2 - 5x - 6\), we look for two numbers that multiply to -6 and add to -5. These numbers are -6 and 1.
- Thus, we can factor the equation as:
\[ (x - 6)(x + 1) = 0 \]

6. **Finding the Roots:**
- Setting each factor to zero gives us the roots:
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Conclusion:
The correct roots of the quadratic equation are \(6\) and \(-1\).