If the two solutions of `x^2-9x+c=0` are complex conjugates, which of the following describes all possible value of c?

Options

A. c=0
B. `cne0`
C. clt9
D. `cgt(81)/4`

Explanation

To determine the values of \( c \) for which the solutions of the equation \( x^2 - 9x + c = 0 \) are complex conjugates, we need to analyze the discriminant of the quadratic equation. The discriminant is given by the formula:

\[ D = b^2 - 4ac \]

For the given quadratic equation \( x^2 - 9x + c = 0 \), we identify the coefficients as follows:
- \( a = 1 \)
- \( b = -9 \)
- \( c = c \)

Now, we can substitute these values into the discriminant formula:

\[ D = (-9)^2 - 4 \cdot 1 \cdot c \]

Calculating \( D \):

\[ D = 81 - 4c \]

For the roots to be complex conjugates, the discriminant must be less than zero:

\[ D < 0 \]

This gives us the inequality:

\[ 81 - 4c < 0 \]

Now, we can solve this inequality for \( c \):

1. Rearranging the inequality:

\[ 81 < 4c \]

2. Dividing both sides by 4:

\[ \frac{81}{4} < c \]

Thus, we can conclude that:

\[ c > \frac{81}{4} \]

This means that the values of \( c \) for which the solutions of the equation are complex conjugates are all values greater than \( \frac{81}{4} \).

### Summary of the Solution:
The possible values of \( c \) such that the solutions of the equation \( x^2 - 9x + c = 0 \) are complex conjugates are:

\[ c > \frac{81}{4} \]