Look at $x^{3/2}-9x^{1/2}$. Both terms contain $x^{1/2}$. Try factoring $x^{1/2}$ out of the numerator.

After factoring out $x^{1/2}$, you will see $x-9$. Think of $x$ as $(\sqrt{x})^2$. How can you factor $x-9$ using $\sqrt{x}$ and 3?

Type the original expression as ((x^(3/2)) - 9x^(1/2))/(sqrt(x) - 3) and restrict the domain to x >= 0 and x != 9 (for example, you can add a restriction like {x>=0, x!=9} in Desmos).

On new lines, enter y = x + 3sqrt(x), y = x - 3sqrt(x), y = sqrt(x) + 3, and y = x - 9. These represent the four choices.

Either compare the graphs visually (which one lies exactly on top of the original expression’s graph for $x \ge 0$, $x \ne 9$?) or pick a value like $x=4$ or $x=16$ (not 9), and use a table or point evaluation to see which choice has the same output as the original expression for that $x$.

Notice that $x^{1/2} = \sqrt{x}$ and $x^{3/2} = (x^{1/2})^3 = (\sqrt{x})^3$.

So the original expression

can be thought of in terms of powers of $\sqrt{x}$.

Factor out the common factor $x^{1/2}$ (which is $\sqrt{x}$) from the numerator:

So the whole fraction becomes

Recognize that $x-9$ is a difference of squares:

• $x = (\sqrt{x})^2$
• $9 = 3^2$

So

Substitute this into the fraction:

Since $x \ne 9$, $\sqrt{x} \ne 3$, so you can safely cancel the common factor $\sqrt{x}-3$ from numerator and denominator.

After canceling $\sqrt{x}-3$, you are left with

Distribute $\sqrt{x}$:

So the expression is equivalent to $x+3\sqrt{x}$, which matches answer choice D.