To find when the business breaks even, set the profit equal to $0$:

Divide both sides by $-2$:

Now factor:

So the solutions are $x=10$ and $x=50$. That means the business breaks even when it sells $10$ units or $50$ units.

A quadratic equation has exactly one solution when its discriminant is $0$. For

the discriminant is

Set this equal to $0$:

Solving gives

and

Factor the numerator:

Then

provided that $x\ne 4$. Since the question asks for an equivalent expression for all values of $x$ except one, the expression is equivalent to $x+4$ for every value in its domain.

A quadratic with vertex $(h,k)$ can be written in vertex form as $f(x)=a(x-h{)}^2+k$. Since the vertex is $(3,-4)$, the function must have the form $f(x)=a(x-3{)}^2-4$. Use the point $(1,4)$ to find $a$:

So the equation is

Set the two height expressions equal because the ball and the top of the wall are at the same height at the same time:

Rearranging gives

Divide by $-8$:

Using the quadratic formula,

The positive solution is about $1.82$, which is closest to $2$. So the correct answer is $2$.