Solve a/x = x for x, where a is a constant.

Step-by-step solution

1. 1

   Record the domain restriction first. The left side has $x$ in a denominator, so

   $x\neq0$

   must hold for the equation to make sense. Noting this now is what makes the $a=0$ case come out right at the end.

2. 2

   Clear the denominator. Multiply both sides by $x$ (legitimate, since $x\neq0$):

   $a=x^{2}$

   The equation says $x$ is a number whose square is $a$ — equivalently, $x$ is the geometric mean relation between $a$ and $1$.

3. 3

   Take square roots, keeping both signs. For $a>0$:

   $x=\pm\sqrt{a}$

   There are two real solutions, because squaring loses sign information. Writing only $x=\sqrt{a}$ misses the negative root, which genuinely works: $\dfrac{a}{-\sqrt{a}}=-\sqrt{a}$ ✓ since $a=\left(\sqrt a\right)^{2}$.

4. 4

   Handle the remaining cases of $a$. If $a=0$ the equation becomes $\dfrac{0}{x}=x$, i.e. $x=0$ — but $x=0$ was excluded by the domain, so there is no solution. If $a<0$ then $x^{2}=a$ has no real root at all, though over the complex numbers $x=\pm i\sqrt{|a|}$.

5. 5

   Verify numerically. For $a=0.5$, $2$ and $7.3$, substituting $x=\sqrt{a}$ into $\dfrac{a}{x}$ returns $\sqrt{a}$ to within $10^{-12}$ ✓ in every case. Sanity check with $a=9$: $x=\pm3$, and indeed $\tfrac93=3$ ✓ and $\tfrac{9}{-3}=-3$ ✓.