Perfect Square Expansion

Lesson focus

A square with side length $a+b$ can be cut into four smaller regions. Their areas explain every term in a perfect square expansion.

A square split into four pieces

Start with a square whose side is $a+b$. Split each side into lengths $a$ and $b$

$a^2$

$ab$

$ab$

$b^2$

$a$$b$$a$$b$

$(a+b)^2=a^2+2ab+b^2$

There are two equal $ab$ rectangles, so together they make $2ab$

Worked example

Expand a perfect square

Expand $(x+5)^2$

Step 1

Square the first term: $x^2$

Step 2

Double the product: $2×x×5=10x$

Step 3

Square the second term: $5^2=25$

Step 4

Combine the three parts: $(x+5)^2=x^2+10x+25$

When the sign is negative

For $(a-b)^2$, algebraic multiplication introduces two products involving $-b$, so the middle term is negative.

$(a-b)^2=a^2-2ab+b^2$

For example, $(x-4)^2=x^2-8x+16$