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

Options

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

Answer

To find the correct roots of the quadratic equation based on the information provided, we can follow these steps:

### Step 1: Identify the incorrect roots and their implications
The first student found the roots to be 3 and 2. From Vieta's formulas, we know that for a quadratic equation of the form \( ax^2 + bx + c = 0 \):
- The sum of the roots \( r_1 + r_2 = -\frac{b}{a} \)
- The product of the roots \( r_1 \cdot r_2 = \frac{c}{a} \)

For the roots 3 and 2:
- Sum of roots: \( 3 + 2 = 5 \)
- Product of roots: \( 3 \cdot 2 = 6 \)

### Step 2: Formulate the incorrect quadratic equation
Let’s assume the incorrect equation is:
\[ x^2 + ax + b = 0 \]
From the roots, we can deduce:
- \( b = 6 \) (product of roots)
- \( a = -5 \) (sum of roots)

Thus, the incorrect equation is:
\[ x^2 - 5x + 6 = 0 \]

### Step 3: Identify the correct equation
The second student copied the constant term and the coefficient of \( x^2 \) as -6 and 1 respectively. This means the correct equation can be represented as:
\[ x^2 + ax - 6 = 0 \]
From the previous step, we know \( a = -5 \). Therefore, the correct equation is:
\[ x^2 - 5x - 6 = 0 \]

### Step 4: Factor the correct quadratic equation
Now, we need to factor the equation \( x^2 - 5x - 6 = 0 \):
\[ x^2 - 5x - 6 = (x - 6)(x + 1) = 0 \]

### Step 5: Solve for the roots
Setting each factor to zero gives us:
1. \( x - 6 = 0 \) → \( x = 6 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

Thus, the correct roots of the quadratic equation are:
\[ \boxed{6 \text{ and } -1} \]

---