E2.5 - Rearranging powers and repeated subjects

A subject inside a power or root

If a square has area $25\,\mathrm{cm}^2$, why is its side five centimetres when both $5^2$ and $(-5)^2$ equal twenty-five? Algebra finds possible values; the meaning of a length selects the positive one.

Squaring can hide an input's sign: opposite inputs have the same square. Thus $x^2=k$ gives $x=\pm\sqrt{k}$ for $k>0$, one value $x=0$ for $k=0$, and no real value for $k<0$. The symbol $\sqrt{k}$ itself means the non-negative square root; the separate $\pm$ gives the two branches.

Keep exactly the same formula but restrict $x$ to a non-negative quantity. The algebra is unchanged until the final branch choice.

When the target starts inside a square root, isolate that root before squaring. For example,

Retain $y\ge0$. Without it, $y=-3$ would give $x=5$, but the original right side is $\sqrt9=3$, not $-3$. For $y=3$, $x=5$ does satisfy the original.

An odd power behaves differently: $x^3=k$ has the single real solution $x=\sqrt[3]{k}$, including negative $k$. There is no extra $\pm$ for a cube root.

A restriction on the target belongs beside the rearranged formula as well as beside any numerical answer.

Collecting a repeated subject

When the target appears twice, undoing operations on only one occurrence cannot isolate it. Bring its terms together first. Then use a common factor: $bx-2x=(b-2)x$. This is the distributive law used backwards.

Leave all terms without the target on the other side, even when those terms contain other letters. They are constants relative to the target you are isolating.

Change only $+2x$ to $-2x$ on the left. Adding $2x$ now creates a sum inside the factorised coefficient.