Two candidates attempt to solve a quadratic equation of the form `x^(2)+px+q=0`. One starts with a wrong value of p and finds the roots to be 2 and 6. The other starts with a wrong value of q and finds the roots to be 2,-9.The correct roots are

Options

A. 3,4
B. 5,3
C. `-3.-4`
D. none

Answer

To solve the problem, we need to determine the correct roots of the quadratic equation \(x^2 + px + q = 0\) based on the information given about the two candidates.

### Step 1: Analyze the first candidate's roots
The first candidate found the roots to be 2 and 6.

Using the properties of roots:
- **Sum of roots**: \( \alpha + \beta = 2 + 6 = 8 \)
- **Product of roots**: \( \alpha \cdot \beta = 2 \cdot 6 = 12 \)

From the quadratic equation, we know:
- The sum of the roots is given by \( -p \), so:
\[ -p = 8 \implies p = -8 \]
- The product of the roots is given by \( q \), so:
\[ q = 12 \]

### Step 2: Analyze the second candidate's roots
The second candidate found the roots to be 2 and -9.

Again, using the properties of roots:
- **Sum of roots**: \( \alpha + \beta = 2 + (-9) = -7 \)
- **Product of roots**: \( \alpha \cdot \beta = 2 \cdot (-9) = -18 \)

From the quadratic equation, we know:
- The sum of the roots is given by \( -p \), so:
\[ -p = -7 \implies p = 7 \]
- The product of the roots is given by \( q \), so:
\[ q = -18 \]

### Step 3: Determine the correct values of \(p\) and \(q\)
We have two different values for \(p\) based on the two candidates:
1. From the first candidate: \( p = -8 \) and \( q = 12 \)
2. From the second candidate: \( p = 7 \) and \( q = -18 \)

Since both candidates started with incorrect values, we need to find a consistent solution for \(p\) and \(q\) that satisfies both conditions.

### Step 4: Find the correct roots
We can use the known product and sum of the roots from both candidates to find the correct roots.

Let’s denote the correct roots as \(r_1\) and \(r_2\):
- From the first candidate, we have:
\[ r_1 + r_2 = -p \quad \text{(unknown)} \]
\[ r_1 \cdot r_2 = q \quad \text{(unknown)} \]

- From the second candidate:
\[ r_1 + r_2 = -(-7) = 7 \]
\[ r_1 \cdot r_2 = -18 \]

Now we can form the quadratic equation using the sum and product of the roots:
\[ x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0 \]
Substituting the values:
\[ x^2 - 7x - 18 = 0 \]

### Step 5: Solve the quadratic equation
To find the roots, we can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \(a = 1\), \(b = -7\), and \(c = -18\):
\[ x = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-18)}}{2 \cdot 1} \]
\[ x = \frac{7 \pm \sqrt{49 + 72}}{2} \]
\[ x = \frac{7 \pm \sqrt{121}}{2} \]
\[ x = \frac{7 \pm 11}{2} \]

Calculating the two possible values:
1. \(x = \frac{18}{2} = 9\)
2. \(x = \frac{-4}{2} = -2\)

### Conclusion
The correct roots of the quadratic equation are \(9\) and \(-2\).