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# Square of a Binomial

## Introduction

The square of a binomial is an essential concept in algebra, where students learn to expand and simplify algebraic expressions. This article provides a comprehensive understanding of this topic, including its definition, key concepts, and illustrative examples. Additionally, it will discuss its real-life applications and provide practice tests and frequently asked questions to help reinforce the knowledge.

The square of a binomial is typically introduced in middle school (usually around 7th or 8th grade) and is further explored in high school algebra classes.

## Math Domain

The square of a binomial falls under the domain of Algebra, specifically focusing on polynomial expressions and their operations.

## Applicable Common Core Standards

This topic aligns with the Common Core State Standards for Mathematics, particularly the standards for Expressions and Equations in Grade 8 and Algebra I.

## Definition of the Topic

The square of a binomial is the result of multiplying a binomial expression by itself. A binomial expression consists of two terms, usually in the form (a + b) or (a – b). When squaring a binomial, we follow the distributive property, resulting in a trinomial expression.

## Key Concepts

Binomial: A binomial has two terms connected by a plus or minus sign.

Square/Squaring: Square means the product of a number when you multiply it by itself. For example, 9 is the square of 3. Squaring means when multiplying a number by itself. For example, the square of 6 is 36.

Perfect Square Trinomials: Perfect square trinomials are polynomials with three terms. It is the result when you square a binomial.

Sum of Squares: (a + b)2= a2 + 2ab + b2

Difference of Squares: (a – b)2 = a2 – 2ab + b2

## Discussion with Illustrative Examples

The square of a binomial is a type of special product where you always end up with a perfect square trinomial as the answer.

Squaring a binomial can be done by:

1. Using the distributive property

(a+b)2 = a(a+b) + b(a+b)
(a-b)2 = a(a-b) – b(a-b)

2. Using the formulas

(a+b)2 = a2 + 2ab + b2
(a-b)2 = a2 – 2ab + b2

Example 1

Find the square of x + 3.

Solution

Since we must square the binomial x+3, we must enclose it in parentheses and use the proper exponent, raising the second power.

Using the distributive property, we shall have this calculation:

(x + 3)2

= (x + 3) (x + 3)

=x(x+3) + 3(x+3)

= x2 + 3x + 3x + 9

= x2 + 6x + 9

Using the formula:

(x + 3)2

=x2+2(x)(3)+32

=x2+6x+9

Notice that we arrive at the same answer, and using the formula is a shortcut way of calculating the square of a binomial without applying the distributive property.

Example 2

Find the square of (2y – 5).

Solution

Using the distributive property to get the product we have,

(2y – 5)2

= (2y – 5)(2y – 5)

= 2y(2y – 5) – 5(2y – 5)

= 4y2 – 10y – 10y + 25

= 4y2 – 20y + 25

Using the formula:

(2y – 5)2

=(2y)2 – 2(2y)(5) + 52

=4y2 – 20y + 25

## Examples with Solution

Example 1

Find the square of the binomial 5x+y using the distributive property.

Solution

(5x+ y)2

= (5x+ y)(5x+ y)

= 5x(5x+y) – y(5x+y)

= 25x2 + 5xy + 5xy + y2

= 25x2 + 10xy + y2

Example 2

Find (3x-4y)2 using the formula.

Solution

a=3x while b=4y. Applying the formula we have,

(3x-4y)2

=(3x)2 – 2(3x)(4y) + (4y)2

=9x2 – 24xy + 16y2

Example 3

Using (a+b)2, find the value of (205)2.

Solution

Since we know that (a+b)2=a2+2ab+b2, we can have a=200 and b=5.

Hence, (205)2=(200+5)2.

(200+5)2

=(200)2+2(200)(5)+52

=40000+2000+25

=42025

## Real-life Application with Solution

Problem 1

A square garden has a side length of (x + 2) meters. Find the area of the garden in terms of x.

Solution

A square’s area is equal to the square of its side length.

Area = (x + 2)2

= (x + 2)(x + 2)

= x2 + 2x + 2x + 4

= x2 + 4x + 4

Therefore, the area of the square garden with a side length of x+2 is x2+4x+4 square meters.

Problem 2

Susan is making a square frame with side lengths of 5-3x. Calculate its area.

Solution

Area = (5-3x)2

= (5-3x)(5-3x)

= 25 – 15x – 15x + 9x2

= 25 – 30x + 9x2

The area of the square frame is 9x2 – 30x + 25 square units.

## Practice Test

1. Find the square of (p + 4).

2. Find the square of (3q – 7).

3. Find the square of (5 – m).

4. Find the square of (3z + 1).

5. Find the square of (t – 6).

6. Evaluate (55)2 using a square of a binomial.

7. Find the area of a square with side lengths (4m-2n) meters.

1. p2 + 2(p)(4) + (4)2 = p2 + 8p + 16

2. (3q)2 – 2(3q)(7) + (7)2 = 9q2 – 42q + 49

3. (5)2 – 2(5)(m) + m2 = 25 – 10m + m2

4. (3z)2 + 2(3z)(1) + (1)2 = 9z2 + 6z + 1

5. t2 – 2(t)(6) + (6)2 = t2 – 12t + 36

6. (50+5)2 = (50)2 + 2(50)(5) + (5)2 = 2500 + 500 + 25 = 3025

7. (4m-2n)2 = (4m)2 – 2(4m)(2n) + (2n)2 = 16m2 – 16mn + 4n   square meters

### Why do we need to learn the square of a binomial?

Learning the square of a binomial is essential for understanding more complex algebraic concepts, such as factoring, simplifying expressions, and solving quadratic equations.

### Can the square of a binomial be negative?

The square of a binomial will always be non-negative because squaring any real number results in a non-negative value.

### What is the difference between squaring a binomial and squaring a monomial?

Squaring a binomial involves multiplying a binomial expression (with two terms) by itself, resulting in a trinomial expression. On the other hand, squaring a monomial (a single term) involves multiplying the monomial by itself, resulting in another monomial with an exponent that is twice the original.

### Is there a shortcut to squaring a binomial without using the distributive property?

Yes, you can use the formulas (a + b)2= a2 + 2ab + b2 and (a – b)2 = a2 – 2ab + b2 as shortcuts to square binomials without applying the distributive property.