Question

find the discriminant.
$3 = -2z^{2} + 2z$
how many real solutions does the equation have?
no real solutions
one real solution
two real solutions

Explanation:

Step1: Rearrange to standard form

Rearrange $3 = -2z^2 + 2z$ to $-2z^2 + 2z - 3 = 0$, or multiply by -1: $2z^2 - 2z + 3 = 0$

Step2: Identify coefficients

For $az^2 + bz + c = 0$, $a=2$, $b=-2$, $c=3$

Step3: Calculate discriminant

Use formula $\Delta = b^2 - 4ac$
$\Delta = (-2)^2 - 4(2)(3) = 4 - 24 = -20$

Step4: Analyze discriminant sign

Since $\Delta < 0$, no real roots exist.

Answer:

Discriminant: $\boldsymbol{-20}$
Number of real solutions: no real solutions