Factoring Special Products in Difference of Squares

Factoring Special Products in

Difference of Squares

By L.D.

Problem 1

9x2-4

Problem 1

9x2 - 4

3x(3x) 2(2)

Both of these can be squared so I will show you what they are squared by under them. To make our problem we will try to fit the formula a2 + b2 = (a + b)(a-b). Since the first one is a2 (9x2), a is 3x, using this way of thinking, I would say that b is 2. Our answer is on the next page.

Problem 1

9x2 – 4 = (3x + 2)(3x – 2)

Example Problems

a2 – 81

36m2 – 25

4x2 – y2

a2 + 64

Remember that both must be PERFECT squares.

Example Problems

a2 – 81 (a + 9)(a – 9)

36m2 – 25 (6m + 5)(6m – 9)

4x2 – y2 (2x + y)(2x – y)

a2 + 64

This cannot be factored since this method doesn’t work with addition problems, only subtraction.

Mini Lesson

If you feel you are just doing the problems blindly, check them with F.O.I.L. and you will find that two of the numbers cancel out together.

Problem 2

2a2 – 200

Problem 2

2a2 – 200

To make this problem work so that we have squares we will have to divide it by 2.

a2 – 100

Now we can solve that to get (a + 10)(a – 10). We add the 2 back by placing it next to the problem for multiplication, making our final answer look like the slide on the next page.

Problem 2

2a2 – 200 = 2(a + 10)(a – 10)

Problem 3

-4c2 + 36

Problem 3

-4c2 + 36

To make this work we will remember what we did in the last problem and divide the problem by -4, making it c2 - 9. Solving this the normal way we will get

(c + 3)(c-3) which will change to be -4(c + 3)(c-3).

Problem 3

-4c2 + 36 = -4(c + 3)(c-3)

Formula

a2 + 2ab + b2 = (a + b)(a + b) or (a + b)2

Problem 4

25x2+ 10x + 1

Problem 4

25x2+ 10x + 1

We will answer this using the formula on slide 13.

a2 + 2ab + b2 = (a + b)(a + b) or (a + b)2

25x2+ 10x + 1

We will first get the square root the 25x2 (5x) and place it in the ‘a’ place of the formula. Then we will get the square root of 1 (1) and place it in the ‘b’ place of the formula. The answer will be on the next slide.

Problem 4

25x2+ 10x + 1 = (5x + 1)2

Problem 5

144y2 - 120y + 25

Problem 5

144y2 - 120y + 25

We need to find a way to accommodate the negative sign in the middle so just blindly using our formula to get (12y + 5)(12y + 5) won’t work. We can however, make it (12y - 5)(12y – 5), which will achieve our goals perfectly.

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