Question

Find the discriminant.

5z^2 - 7z - 9 = 0
How many real solutions does the equation have?

Answer 1

To find the discriminant of a quadratic equation in the form \( az^2 + bz + c = 0 \), we use the formula:

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

For the given quadratic equation \( 5z^2 - 7z - 9 = 0 \), the coefficients are:

- \( a = 5 \)
- \( b = -7 \)
- \( c = -9 \)

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

\[
D = (-7)^2 - 4(5)(-9)
\]

Calculating \( (-7)^2 \):

\[
(-7)^2 = 49
\]

Now calculate \( 4(5)(-9) \):

\[
4(5)(-9) = -180
\]

Substituting these values back into the discriminant expression:

\[
D = 49 - (-180)
\]
\[
D = 49 + 180
\]
\[
D = 229
\]

Since the discriminant \( D = 229 \) is greater than zero, this indicates that the quadratic equation has **two distinct real solutions**.