Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 11: Special Factorizations

Lesson 1: Squares of Binomials

In this Grade 4 AMC Math lesson from AoPS: Introduction to Algebra, students learn how to expand and factor squares of binomials using the identity (a + b)² = a² + 2ab + b². The lesson covers recognizing perfect square trinomials, applying the pattern to both positive and negative terms, and factoring expressions back into binomial squares. Students practice identifying whether a quadratic expression fits the perfect square form across a range of problems, including those with fractions and two variables.

Section 1

Squares of Binomials

Property

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

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

Examples

To expand $(x + 5)^2$ , we use the formula with $a=x$ and $b=5$ : $(x)^2 + 2(x)(5) + (5)^2$ , which simplifies to $x^2 + 10x + 25$ .

Section 2

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

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

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

To use this pattern, first verify that the trinomial fits.
Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

).
Then, check if the middle term is twice their product (

2ab

).
If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Squares of Binomials

Property

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

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

Examples

To expand $(x + 5)^2$ , we use the formula with $a=x$ and $b=5$ : $(x)^2 + 2(x)(5) + (5)^2$ , which simplifies to $x^2 + 10x + 25$ .

Section 2

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

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

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

To use this pattern, first verify that the trinomial fits.
Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

).
Then, check if the middle term is twice their product (

2ab

).
If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Squares of Binomials

Property

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

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

Examples

To expand $(x + 5)^2$ , we use the formula with $a=x$ and $b=5$ : $(x)^2 + 2(x)(5) + (5)^2$ , which simplifies to $x^2 + 10x + 25$ .

Section 2

Factor Perfect Square Trinomials

Property

If $a$ and $b$ are real numbers, the perfect square trinomials pattern is as follows:

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

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

To use this pattern, first verify that the trinomial fits.
Check if the first term is a perfect square (

a^2

) and the last term is a perfect square (

b^2

).
Then, check if the middle term is twice their product (

2ab

).
If it matches, write the square of the binomial

(a+b)^2

(a-b)^2

Examples

To factor $25x^2 + 30x + 9$ , recognize it as $(5x)^2 + 2(5x)(3) + 3^2$ . This fits the pattern $a^2+2ab+b^2$ , so the factored form is $(5x+3)^2$ .

To factor $49y^2 - 42y + 9$ , identify it as $(7y)^2 - 2(7y)(3) + 3^2$ . This matches the pattern $a^2-2ab+b^2$ , so the factored form is $(7y-3)^2$ .