Question

Find the discriminant.

z^2 - 6z + 9 = 0
How many real solutions does the equation have?

Answer 1

To find the discriminant of the quadratic equation given in the form \( z^2 - 6z + 9 = 0 \), we use the formula for the discriminant \( D \):

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

In this equation, the coefficients are:
- \( a = 1 \) (the coefficient of \( z^2 \)),
- \( b = -6 \) (the coefficient of \( z \)),
- \( c = 9 \) (the constant term).

Now, substitute these values into the discriminant formula:

\[
D = (-6)^2 - 4(1)(9)
\]

Calculating \( D \):

\[
D = 36 - 36 = 0
\]

The discriminant \( D = 0 \) indicates that there is exactly one real solution to the equation.

Thus, the number of real solutions is **1**.