Question

what is the value of the discriminant, $b^{2}-4ac$, for the quadratic equation $0 = -2x^{2}-3x + 8$, and what does it mean about the number of real solutions the equation has?
the discriminant is -55, so the equation has 2 real solutions.
the discriminant is -55, so the equation has no real solutions.
the discriminant is 73, so the equation has 2 real solutions.
the discriminant is 73, so the equation has no real solutions.

Explanation:

Step1: Identify a, b, c values

For the quadratic equation $0=-2x^{2}-3x + 8$, we have $a=-2$, $b=-3$, $c = 8$.

Step2: Calculate the discriminant

Substitute the values into the discriminant formula $b^{2}-4ac$.
\[

$$\begin{align*} (-3)^{2}-4\times(-2)\times8&=9+64\\ &=73 \end{align*}$$

\]

Step3: Determine the number of real - solutions

If the discriminant $\Delta=b^{2}-4ac>0$, the quadratic equation has 2 real solutions. Since $\Delta = 73>0$, the equation has 2 real solutions.

Answer:

The discriminant is 73, so the equation has 2 real solutions.