Answer
Discriminant: $
32
$
Number of Real Solutions: $
2
$
Number of Imaginary Solutions: $
0
$
Work Step by Step
Using $ax^2+bx+c=0,$ the given equation,
\begin{align*}
-4x^2-4x+1=0
,\end{align*} has $a=
-4
,$ $b=
-4
,$ and $c=
1
.$ Using $b^2-4ac$ or the Discriminant, then
\begin{align*}\require{cancel}b^2-4ac&\Rightarrow
(-4)^2-4(-4)(1)
\\&=
16+16
\\&=
32
.\end{align*}
Since the value of the discriminant is greater than $0,$ then there are $2$ real solutions.