Skill: Expand special products of binomials

Special products are shortcuts based on the distributive property. The three important identities are

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

and

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

The middle term in a squared binomial is doubled because the two cross-products are alike. In conjugate products, the cross-products cancel.

Example: Expand $((2x-3)^2)$

Recognize this as the pattern

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

Here,

$$a=2x \qquad \text{and} \qquad b=3.$$

Substitute these values into the identity:

$$(2x-3)^2=(2x)^2-2(2x)(3)+3^2.$$

Now simplify each term:

$$(2x)^2=4x^2,$$ $$-2(2x)(3)=-12x,$$

and

$$3^2=9.$$

Therefore,

$$\boxed{(2x-3)^2=4x^2-12x+9}.$$

Be careful not to forget the middle term. For a squared binomial, it is always twice the product of the two terms, with the sign determined by the original binomial.