**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.

**Grade Appropriateness**

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}= a

^{2}+ 2ab + b

^{2}

** Difference of Squares**: (a – b)

^{2}= a

^{2}– 2ab + b

^{2}

**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} = a^{2} + 2ab + b^{2}

(a-b)^{2} = a^{2} – 2ab + b^{2}

**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)

= x^{2} + 3x + 3x + 9

= x^{2} + 6x + 9

Using the formula:

(x + 3)^{2}

=x^{2}+2(x)(3)+3^{2}

=x^{2}+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)

= 4y^{2} – 10y – 10y + 25

= 4y^{2} – 20y + 25

Using the formula:

(2y – 5)^{2}

=(2y)^{2} – 2(2y)(5) + 5^{2}

=4y^{2} – 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)

= 25x^{2} + 5xy + 5xy + y^{2}

= 25x^{2} + 10xy + y^{2}

**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}

=9x^{2} – 24xy + 16y^{2}

**Example 3**

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

**Solution**

Since we know that (a+b)^{2}=a^{2}+2ab+b^{2}, we can have a=200 and b=5.

Hence, (205)^{2}=(200+5)^{2}.

(200+5)^{2}

=(200)^{2}+2(200)(5)+5^{2}

=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)

= x^{2} + 2x + 2x + 4

= x^{2} + 4x + 4** **

Therefore, the area of the square garden with a side length of x+2 is **x ^{2}+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 + 9x^{2}

= 25 – 30x + 9x^{2}

The area of the square frame is 9x^{2} – 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.

*Answers:*

1. p^{2} + 2(p)(4) + (4)^{2} = p^{2} + 8p + 16

2. (3q)^{2} – 2(3q)(7) + (7)^{2} = 9q^{2} – 42q + 49

3. (5)^{2} – 2(5)(m) + m^{2} = 25 – 10m + m^{2}

4. (3z)^{2} + 2(3z)(1) + (1)^{2} = 9z^{2} + 6z + 1

5. t^{2} – 2(t)(6) + (6)^{2} = t^{2} – 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} = 16m^{2} – 16mn + 4n square meters

**Frequently Asked Questions (FAQs)**

**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}= a^{2} + 2ab + b^{2} and (a – b)^{2} = a^{2} – 2ab + b^{2} as shortcuts to square binomials without applying the distributive property.

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