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
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