After you combine the fractions, the numerator and the expression inside the square root will match. What happens if you let $t = x^2 - a^2$ and simplify $\dfrac{t}{\sqrt{t}}$?

Once you have an equation involving $\sqrt{x^2 - a^2}$, how can you remove the square root? After that, how do you solve for $x^2$ and then for $x$?

You should end up with two possible $x$ values. Look carefully at each answer choice and see which one matches either of the values you found.

In Desmos, type a = 3 (or any positive value for $a$) to create a slider. Then enter the left side as y = x^2/sqrt(x^2-a^2) - a^2/sqrt(x^2-a^2) and also enter the horizontal line y = 52.

Use Desmos’s intersection tool (click where the graphs cross or use the menu) to find the $x$-coordinates where the curve $y = \dfrac{x^2}{\sqrt{x^2-a^2}} - \dfrac{a^2}{\sqrt{x^2-a^2}}$ intersects the line $y = 52$.

Take one of the intersection $x$-values and create a new expression x^2 - (a^2 + 52^2) in Desmos. For different positive values of $a$, you should see this expression stay very close to 0, confirming that each solution satisfies $x^2 = a^2 + 52^2$.

Start with the equation

Both fractions have the same denominator $\sqrt{x^2 - a^2}$, so combine them:

Also note that $x^2 - a^2$ must be positive so that the square root and denominator are defined.

Let $t = x^2 - a^2$ (so $t>0$).

Then the equation becomes

For $t>0$, we have $\dfrac{t}{\sqrt{t}} = \sqrt{t}$, so this simplifies to

Now you have a basic square root equation.

Square both sides of

to eliminate the square root:

Now solve for $x^2$ by adding $a^2$ to both sides:

From

take the square root of both sides:

Both signs are valid solutions, because they give the same positive value for $x^2 - a^2$, so they satisfy the original equation. Among the answer choices, the only one that matches one of these solutions is $-\sqrt{a^2 + 52^2}$, which is choice C.