In finding the roots of `x^(2) + px + q = 0` the coefficient of x was taken as -7 instead of -8 and the roots were 4 and 3 then the correct roots are

Options

A. 4, 3
B. 2, 4
C. 2, 6
D. 4, 6

Answer

To solve the problem step by step, we need to find the correct roots of the quadratic equation given that the coefficient of \( x \) was incorrectly taken as -7 instead of -8.

### Step 1: Identify the given roots and their properties
The roots provided are 4 and 3. We can use these roots to find the values of \( p \) and \( q \) in the quadratic equation \( x^2 + px + q = 0 \).

**Hint:** The sum of the roots \( r_1 + r_2 = -p \) and the product of the roots \( r_1 \cdot r_2 = q \).

### Step 2: Calculate the sum and product of the roots
- The sum of the roots \( 4 + 3 = 7 \).
- The product of the roots \( 4 \cdot 3 = 12 \).

Since the coefficient of \( x \) was incorrectly taken as -7, we need to find the correct values:
- The correct sum of the roots should be \( -p = -8 \) (as per the problem statement), which gives \( p = -8 \).
- The product \( q = 12 \).

### Step 3: Write the correct quadratic equation
Using the values of \( p \) and \( q \), we can write the correct quadratic equation:
\[ x^2 - 8x + 12 = 0 \]

**Hint:** The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \).

### Step 4: Factor the quadratic equation
Now, we need to factor the equation \( x^2 - 8x + 12 = 0 \):
\[ x^2 - 8x + 12 = (x - 6)(x - 2) = 0 \]

**Hint:** To factor, look for two numbers that multiply to \( c \) (12) and add to \( b \) (-8).

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

Thus, the correct roots of the equation are \( 6 \) and \( 2 \).

### Final Answer
The correct roots are **2 and 6**.