Adding and Subtracting a Term

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Adding zero in a useful form can reveal a square or another factorable pattern. The expression is unchanged only when the added and subtracted terms are exactly equal.

Visual guide

VISUAL GUIDEMake the hidden square visible

(x² + 2)²
(2x)²
Their difference: x⁴ + 4

Adding and subtracting 2x² turns x⁴ + 4 into (x² + 2)² − (2x)². The blue curve is the larger square; subtracting the orange square leaves a positive gap of at least 4.

Method

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

x^2+bx+c=(x+b/2)^2+c-b^2/4.

This also identifies the vertex of its graph.

Worked example

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

x^4+4=(x^2+2)^2-(2x)^2.

A difference of squares now gives

(x^2-2x+2)(x^2+2x+2).

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

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which added-and-subtracted term helps factor

x^4+4

4x²

Hint 1 · Find a starting point

Try to create (x²+2)².

Hint 2 · Take the next step

That square equals x⁴+4x²+4.

Show the reasoning

Answer: 4x²

x⁴+4 = (x²+2)²−(2x)² = (x²−2x+2)(x²+2x+2).

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

Common mistake

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

Check your understanding

Complete the square in $x^2+6x+5$ .

Show answer

$(x+3)^2-4$ , so it also factors as $(x+1)(x+5)$ .

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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