If one root is `3-sqrt5` and the other root is `3+sqrt5`, find the possible sum of coefficients of `x^2` and x.

Options

A. `-20`
B. `-5`
C. 14
D. 11

Answer

To solve the problem, we need to find the sum of the coefficients of \(x^2\) and \(x\) in the quadratic equation formed by the given roots \(3 - \sqrt{5}\) and \(3 + \sqrt{5}\).

### Step-by-Step Solution:

1. **Identify the Roots**:
The roots given are \(r_1 = 3 - \sqrt{5}\) and \(r_2 = 3 + \sqrt{5}\).

2. **Calculate the Sum of the Roots**:
The sum of the roots \(S\) can be calculated as:
\[ S = r_1 + r_2 = (3 - \sqrt{5}) + (3 + \sqrt{5}) = 3 + 3 = 6 \]

3. **Calculate the Product of the Roots**:
The product of the roots \(P\) can be calculated as:
\[ P = r_1 \cdot r_2 = (3 - \sqrt{5})(3 + \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \]

4. **Form the Quadratic Equation**:
Using the sum and product of the roots, the quadratic equation can be formed as:
\[ x^2 - Sx + P = 0 \]
Substituting the values of \(S\) and \(P\):
\[ x^2 - 6x + 4 = 0 \]

5. **Identify the Coefficients**:
The coefficients of the quadratic equation \(x^2 - 6x + 4\) are:
- Coefficient of \(x^2\) (let's call it \(a\)) = 1
- Coefficient of \(x\) (let's call it \(b\)) = -6

6. **Calculate the Sum of the Coefficients**:
Now, we find the sum of the coefficients of \(x^2\) and \(x\):
\[ a + b = 1 + (-6) = 1 - 6 = -5 \]

### Final Answer:
The possible sum of the coefficients of \(x^2\) and \(x\) is \(-5\).