X-Puzzles This is usually introduced in Pre-algebra (7th

grade). It’s a simple pattern that is discovered, but

tends to not be taught.

X-Puzzles

Using the pattern in puzzles A and B, complete puzzles C, D and E.

Did you get?

10

7 1410

21 24

Pretty easy, right?

A. B. C. D. E.

5

6

3

2

7

12

4

3

5

2

3

7

12

2

X-Puzzles

F. G. H. I. J.

5

4

10

2

10

2

7

4

3

2

Try these!

X-Puzzles

Look at these puzzles when different parts are missing

12

6

1

1

5

3

15

8

36

12

X-Puzzles - continued

See if you can solve these harder ones.

X puzzles – WHY WERE WE DOING THIS???

When we did the area models They always were set up for us We knew what the coefficients of x were

We DON’T know how to fill in the blanks if they are missing

The X-Puzzles Give you the coefficients that go in the missing

spaces. Allow you to have a visual for finding unknown values. Are pretty straight forward once you get the hang of

them.

Here’s an example of using the X-PuzzleGiven: x2 + 13x +36

36

9

13

4x2

369x

4xx

9

x 4

Answer: x2 + 13x +36 = (x + 9)(x + 4)

Same numbers that were in the x-puzzle!

You try one! Given: x2 + 5x + 6

6

5

2 3

2x

3xx2

6

x

2

x 3

Answer: x2 + 5x + 6 = (x + 2)(x + 3)

One more… Given: x2 - 10x - 24

Answer: x2 - 10x – 24=(x + 2)(x - 12)

-24

-10

-12 2

-12x

2xx2

-24

x

-12

2x

It even works with the difference of 2 squares (“b” term is missing)! Given: x2-9

-9

0

-3 3

-3x

3xx2

-9

x

-3

x 3

Answer: x2 - 9 = (x - 3)(x + 3)

It also works if the constant term is missing! Given: 4x2-8x

0

-8x

0 -8x

-8x

04x2

0

x

-2

4x 0

Answer: 4x(x - 2)

Since there is a column of zeros, we can get rid of it

Now, it’s time to see you do it on your own Set up the x puzzles and the area models to

factor the following polynomials.

1)x2 + 3x + 22)x2 + 5x + 63)x2 - 7x + 104)x2 - 8x - 9

Now, there are times when you’re given a coefficient in front of the x2. Don’t panic, you still have all the tools

necessary to solve these, we just need to modify our x-puzzles.

Example: 2x2 + 3x + 1 HOWEVER, you’ll need to look at the

coefficient on the 2x2

2x2

1

1x

2x

2*? ?

3

Thus, 2x2+3x+1=(2x+1)(x+1)

2*1 12x

x

1

1

Let’s look at a harder one. 3x2 + 11x + 10 Still, you’ll need to look at the coefficient on

the x2

3x2

10

10x

3x

3*? ?

11

Thus, 3x2+11x+10=(3x+5)(x+2)

3*2 56x

5x

2

5

Here is the modification that is much easier. 3x2 + 11x + 10 ALL YOU NEED TO DO, IS MULTIPLY YOUR

OUTSIDE NUMBERS FIRST. (3x10) This goes on your x-puzzle where the product normally goes.

3x2

10

30x

3x

? ?

11

Thus, 3x2+11x+10=(3x+5)(x+2)

5 66x

5x

2

5

One more, but the steps are broken down. 2x2 + 9x + 10

2x2

10

2x

x

Step 1: Draw the area model and x-puzzle

Step 2: Fill in what information you can in both the area model and x-puzzle.

20

9

Step 3: Multiply the outside numbers. In this case, 2 and 10.

? ?

Step 4: Solve the x-puzzle and put those values into your area model

4x

5x

4 5

2

5

Step 5: Find the missing pieces and write your final answer.

2x2 + 9x + 10 = (2x+5)(x+2)

2x10=20

Try these three on your own

1)4x2 + 4x -32)2x2 + 7x + 53)8x2 – 14x -9

And that’s the basics to factoring with two visual tools