Question

find the discriminant.
$2 = -5z^2 + 3z$
how many real solutions does the equation have?
no real solutions
one real solution
two real solutions

Explanation:

Step1: Rearrange to standard quadratic form

Rearrange $2 = -5z^2 + 3z$ to $az^2 + bz + c = 0$:
$$5z^2 - 3z + 2 = 0$$

Step2: Identify coefficients

From $5z^2 - 3z + 2 = 0$, we get:
$a=5$, $b=-3$, $c=2$

Step3: Calculate discriminant

Use discriminant formula $\Delta = b^2 - 4ac$:
$$\Delta = (-3)^2 - 4(5)(2) = 9 - 40 = -31$$

Step4: Determine real solutions

Since $\Delta < 0$, the quadratic has no real solutions.

Answer:

Discriminant: $\boldsymbol{-31}$
Number of real solutions: $\boldsymbol{\text{no real solutions}}$