Question

Find the discriminant.

9z^2 - 3 = 0
How many real solutions does the equation have?

Answer 1

To find the discriminant of the quadratic equation \( 9z^2 - 3 = 0 \), we first need to rewrite it in the standard form \( az^2 + bz + c = 0 \), where \( a \), \( b \), and \( c \) are constants.

In this case:
- \( a = 9 \)
- \( b = 0 \) (since there is no \( z \) term)
- \( c = -3 \)

The formula for the discriminant \( D \) of a quadratic equation is given by:

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

Now substituting the values of \( a \), \( b \), and \( c \):

\[
D = 0^2 - 4(9)(-3)
\]
\[
D = 0 - (-108)
\]
\[
D = 108
\]

Now that we have calculated the discriminant, we find that \( D = 108 \).

Since the discriminant is positive \( (D > 0) \), the equation \( 9z^2 - 3 = 0 \) has **two distinct real solutions**.