Adding and Subtracting a Term — Cheat sheet

Key method

Complete a square by supplying the missing middle term, then treat the remainder separately. For a monic quadratic:

This also identifies the vertex of its graph.

Example

To factor $x^4+4$, add and subtract $4x^2$:

A difference of squares now gives

Both factors are positive for real $x$, since they equal $(x\mp1)^2+1$.

Avoid this mistake

Adding a term alone changes the problem. Write the compensating subtraction on the same line.