Sachin and Rahul attempted to solve a quadratic equation. Sachin made a mistake in writing down the constant term and ended up in roots (4,3). Rahul made a mistake in writing down coefficient of x to get roots (3, 2). The correct roots of equation are:

Options

A. `-4,-3`
B. `6,1`
C. `4,3`
D. `-6,-1`

Explanation

To solve the problem step by step, we need to analyze the information given about the roots obtained by Sachin and Rahul, and then derive the correct roots of the original quadratic equation.

### Step 1: Understand the roots obtained by Sachin and Rahul
- Sachin obtained roots \(4\) and \(3\).
- Rahul obtained roots \(3\) and \(2\).

### Step 2: Calculate the sum and product of the roots for both cases
- For Sachin:
- Sum of roots \(S_S = 4 + 3 = 7\)
- Product of roots \(P_S = 4 \times 3 = 12\)

- For Rahul:
- Sum of roots \(S_R = 3 + 2 = 5\)
- Product of roots \(P_R = 3 \times 2 = 6\)

### Step 3: Set up the equations based on the roots
The general form of a quadratic equation based on its roots \(r_1\) and \(r_2\) is:
\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]

For Sachin's case:
\[ x^2 - S_S x + P_S = 0 \implies x^2 - 7x + 12 = 0 \]

For Rahul's case:
\[ x^2 - S_R x + P_R = 0 \implies x^2 - 5x + 6 = 0 \]

### Step 4: Analyze the mistakes
- Sachin made a mistake in the constant term, so the sum of the roots is correct, but the product is wrong.
- Rahul made a mistake in the coefficient of \(x\), so the product of the roots is correct, but the sum is wrong.

### Step 5: Find the correct sum and product
From the analysis:
- The correct sum of the roots is \(S = 7\) (from Sachin).
- The correct product of the roots is \(P = 6\) (from Rahul).

### Step 6: Form the correct quadratic equation
Using the correct sum and product:
\[ x^2 - Sx + P = 0 \implies x^2 - 7x + 6 = 0 \]

### Step 7: Factor the quadratic equation
To factor \(x^2 - 7x + 6\):
\[ x^2 - 7x + 6 = (x - 6)(x - 1) = 0 \]

### Step 8: Solve for the roots
Setting each factor to zero gives:
- \(x - 6 = 0 \implies x = 6\)
- \(x - 1 = 0 \implies x = 1\)

### Final Answer
The correct roots of the quadratic equation are \(1\) and \(6\).

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