## Key Concepts

- Determine the square of a binomial
- Find the product of a sum and a difference
- Apply the square of a binomial

## Determine the Square of a Binomial

### Concept

**Why is (a + b)**^{2}** considered a special case when multiplying polynomials? **

Use the distributive property.

Use a visual model.

The square of a binomial follows the pattern: (a + b)^{2} = a^{2} + 2ab + b^{2}

**Example 1 **

**What is the product (5x – 3)**^{2}**? **

Use the pattern learned in the previous slide to find the square of a difference.

(5x – 3)^{2} = (5x + (-3))^{2} ——————————Rewrite the difference as sum.

= (5x)^{2}+ 2(5x)(-3) + (-3)^{2}————Substitute 5x and -3 into a^{2}+ 2ab + b^{2}

= 25x^{2}– 30x + 9 ————————-Simplify

You can write the product (5x – 3)^{ 2}as 25x^{2}– 30x + 9.

**Example 2**

**How can you use the square of a binomial to find the product 29**^{2}**? **

Rewrite the product as a difference of two values whose squares you know, such as (30 – 1)^{ 2}. Then use the pattern for the square of a binomial to find its square.

So, 29^{2} = 841. In general, you can use the square of a binomial to find the square of a large number by rewriting the number as the sum or difference of two numbers with known squares.

## Find the Product of a Sum and a Difference

### Concept

**What is the product (a + b)(a – b)?**

Use the distributive property to find the product.

The product of two binomials in the form: (a + b)(a – b) is a^{2} – b^{2}.

The product of the sum and the difference of the same two values results in the difference of two squares.

**Example 3**

**What is the product (5x + 7) (5x – 7)?**

Use the pattern learned in the previous slide.

(5x + 7) (5x – 7) = (5x)^{2} – (7)^{2}——— Substitute 5x and 7 into a^{2}– b^{2}.

= 25x^{2}– 49 ———— Simplify.

The product of (5x + 7) (5x – 7) is 25x^{2}– 49. It is the difference of two squares, (5x)^{2} – (7)^{2}.

**Example 4**

**How can you use the difference of two squares to find the product of 43 and 37?**

Rewrite the product as the sum and the difference of the same two numbers, *a* and *b*.

You can use the difference of two squares to mentally find the product of large numbers when the numbers are the same distance from a known square.

### Apply the Square of a Binomial

**Example 5**

A graphic designer is developing images for icons. The square pixelated image is placed inside a border that is 2 pixels wide on all sides. If the area of the border of the image is 176 square pixels, what is the area of the image?

**Solution:**

Let *x* represent the length and the width of the image.

**Formulate: **

The area of the image and the border is represented by the expression (x + 4)^{2}.

**Compute:**

The image will be 20 pixels by 20 pixels. The area of the image is 20*20, or 400 square pixels.

### Questions

**Question 1**

Find each product.

**1. (3x – 4) ^{2} **

**Solution:**

Using the pattern (a + b)^{ 2} = a^{2} + 2ab + b^{2}

(3x + (-4))^{ 2} = (3x)^{ 2} + 2 × 3x × (-4) + (-4)^{ 2} = 9x^{2} – 24x + 16

**2. 71 ^{2} **

**Solution:**

(70 + 1)^{ 2} = (70)^{ 2} + 2 × 70 ×1 + (1)^{ 2} = 4900 + 140 + 1 = 5041

**Question 2**

Find each product.

**1. (2x – 4)(2x + 4) **

**Solution:**

Using the pattern (a + b)(a – b) = a^{2} – b^{2}

(2x – 4)(2x + 4) = (2x)^{ 2} – 4^{2} = 4x^{2} – 16

**2. 56*44 **

**Solution:**

(50 + 6)(50 – 6) = (50)^{ 2} – 6^{2} = 2500 – 36 = 2464

**Question 3**

In example 5, what is the area of the square image if the area of the border is 704 square pixels and the border is 4 pixels wide?

**Solution:**

Let *x* represent the length and the width of the image.

Since the border is 4 pixels wide,

The area of the image and the border is represented by the expression (x + 8)^{2}.

Area of border = total area – area of image

= (x + 8)^{ 2} – x^{2}

= x^{2} + 2 × 8 × x + 8^{2} – x^{2}

= 16x + 64

Since the area of border is 704 square pixels,

16x + 64 = 704

16x = 640

x = 40 pixels

Area of the image = x^{2} = 40^{2} = 1600 square pixels

### Key Concepts Covered

## Exercise

- Find the product of the following equations:
- (x+9) (x-9)
- (x-7) (x-7)
- (2x – 1)
^{ 2} - (x – 7)
^{ 2} - (2x + 5)
^{ 2}

- Kennedy multiplies (x – 3) (x + 3) and gets an answer of x
^{2}– 6x – 9. Describe and correct the Kennedy’s error. - Use the square of a binomial to find the product. 54
^{2} - Explain why the product of two binomials in the form (a + b) (a – b) is a binomial instead of a trinomial.
- Find the product. (3a – 4b) (3a + 4b)
- Find the product. (x
^{2}– 2y) (x^{2}+ 2y)

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