The Difference of Two Squares

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Two squared quantities separated by subtraction factor into a sum and a difference. The cross terms cancel when the two binomials are multiplied. The squared quantities can themselves be expressions.

Method

A^2-B^2=(A-B)(A+B).

Identify both squares explicitly before substituting. This identity is valid for real or complex values.

Worked example

For $9x^2-25$ , use $A=3x$ and $B=5$ to obtain $(3x-5)(3x+5)$ . For $(x+2)^2-9$ , use $A=x+2$ and $B=3$ :

(x+2-3)(x+2+3)=(x-1)(x+5).

This avoids an unnecessary expansion.

Common mistake

The sum $A^2+B^2$ does not equal $(A+B)(A-B)$ . Over the reals, $x^2+1$ has no linear factors.

Check your understanding

Factor $16t^2-49$ .

Show answer

$(4t-7)(4t+7)$ .

Explore

Factors and zeros

Try this. Move the roots together and apart. Match the expanded coefficients to their sum and product, then compare touching with crossing the axis.

First root-2Second root2

(x − (-2))(x − (2)) = x² − (0)x + (-4). Zeros: -2, 2. Simple roots: the graph crosses at each zero.

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.

Factor

x^2-16

(x−4)²

(x−8)(x+2)

(x−4)(x+4)

Hint 1 · Find a starting point

Both terms are squares.

Hint 2 · Take the next step

Use a²−b² = (a−b)(a+b).

Show the reasoning

Answer: (x−4)(x+4)

The cross terms cancel in (x−4)(x+4).

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

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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