Write a/x - x as a single fraction.

Step-by-step solution

1. 1

   Note what kind of task this is. With no equals sign there is nothing to solve — the expression can only be simplified, and for a difference of a fraction and a whole term that means combining over one denominator.

2. 2

   Give the second term the same denominator. The only denominator present is $x$, so rewrite $x$ as a fraction over $x$ by multiplying top and bottom by $x$:

   $x=\frac{x\cdot x}{x}=\frac{x^{2}}{x}$

   This is legal for $x\neq0$, which the original expression already requires.

3. 3

   Subtract the numerators over the shared denominator.

   $\frac{a}{x}-\frac{x^{2}}{x}=\frac{a-x^{2}}{x}$

   Note the whole of $x^{2}$ is subtracted, and the denominator is written once — a frequent error is doubling it to $x^{2}$.

4. 4

   Check whether anything cancels. The numerator $a-x^{2}$ has no factor of $x$ (the $a$ blocks it), so the fraction is fully simplified as it stands. State the restriction $x\neq0$ alongside the answer.

5. 5

   Verify numerically. Comparing $\dfrac{a}{x}-x$ with $\dfrac{a-x^{2}}{x}$ at $(a,x)=(2,1.5)$, $(2,-2.5)$, $(-3,1.5)$ and $(-3,-2.5)$ gives agreement to within $10^{-12}$ in every case ✓. Spot check $a=6$, $x=2$: $3-2=1$ and $\dfrac{6-4}{2}=1$ ✓.

6. 6

   Note the payoff. In this form the equation $\dfrac{a}{x}-x=0$ is transparent: a fraction is zero exactly when its numerator is, so $a-x^{2}=0$ and $x=\pm\sqrt{a}$ — no separate denominator-clearing step is needed.