To find the discriminant of the equation \( 8z^2 + 2 = 9z \), we first need to rearrange it into standard quadratic form \( az^2 + bz + c = 0 \).
Starting with the equation:
\[ 8z^2 - 9z + 2 = 0 \]
Now, we identify the coefficients:
- \( a = 8 \)
- \( b = -9 \)
- \( c = 2 \)
The discriminant \( D \) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
Substituting the values we found:
\[ D = (-9)^2 - 4 \cdot 8 \cdot 2 \] \[ D = 81 - 64 \] \[ D = 17 \]
Now, to determine how many real solutions the equation has, we use the value of the discriminant:
- If \( D > 0 \), there are two distinct real solutions.
- If \( D = 0 \), there is exactly one real solution.
- If \( D < 0 \), there are no real solutions.
Since \( D = 17 \) (which is greater than 0), the equation has two distinct real solutions.