Question 108·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

The radicand is the expression inside a square root. Here its root is also a denominator, so it cannot be zero. That means $x^2-a^2>0$; any solution must satisfy $|x|>a$.

For a sample, set $a=52$, a positive value already in the question. Type the original left side as $g(x)$, then type $g(x_1)\sim52\{x_1<-a\}$. The $\sim$ asks Desmos to solve for $x_1$, and the braces select a negative solution outside the excluded interval. Under PARAMETERS, Desmos gives $x_1\approx-73.54$. This previews one case; it does not determine the formula for every $a$.

Both fractions have the same denominator, so subtract their numerators:

From the first step, $x^2-a^2>0$. A positive number equals its square root multiplied by itself, so dividing that number by its square root leaves one square root. Simplify:

Square both sides of the equation:

Add $a^2$ to both sides:

Taking square roots of $x^2$ gives two branches, because a number and its negative have the same square:

For either branch, $x^2-a^2=52^2>0$. So the original denominator is defined, and its square root is $52$, exactly as the simplified equation requires. The question asks for one solution, so the negative branch works: $-\sqrt{a^2+52^2}$. Choice C.