Lesson 60: Finding Special Products of Binomials

New Concept

What’s next

Next, you’ll apply these patterns to multiply binomials, perform mental math tricks, and solve problems involving area.

Property

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

Explanation

Squaring a binomial means multiplying it by itself, not just squaring the two terms inside! This special pattern is a shortcut so you don't have to use FOIL every time. Just square the first term, square the last term, and find twice their product for the middle. It’s a handy trick for perfect-square trinomials.

Examples

$(x + 5)^2 = x^2 + 2(x)(5) + 5^2 = x^2 + 10x + 25$
$(2x - 4)^2 = (2x)^2 - 2(2x)(4) + 4^2 = 4x^2 - 16x + 16$
$(3t - 1)^2 = (3t)^2 - 2(3t)(1) + 1^2 = 9t^2 - 6t + 1$

Squaring a binomial isn't just squaring each term—let's use the special pattern for squaring binomials.

Example Problem

Find the product: $(3x - 5)^2$.

Property

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

Explanation

When you multiply two binomials that are almost identical—except one has a plus and one has a minus—the middle terms cancel out. This leaves you with a super simple answer: just the first term squared minus the second term squared. It's the ultimate shortcut for these special pairs, resulting in a difference of two squares.

Examples

$(x + 3)(x - 3) = x^2 - 3^2 = x^2 - 9$
$(5x + 4)(5x - 4) = (5x)^2 - 4^2 = 25x^2 - 16$
$(3x + 2)(3x - 2) = (3x)^2 - 2^2 = 9x^2 - 4$

Watch how multiplying a sum and a difference makes the middle term vanish!

Example Problem

Find the product: $(4x + 7)(4x - 7)$.