Let’s solve the problem:
Factor x² + 5x + 6
We’ll use both methods shown — Diamond Method and XBox Method — to factor this trinomial.
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Step 1: Understand what we’re looking for
We want to write
x² + 5x + 6 as two binomials multiplied together, like:
(x + ?)(x + ?)
To do that, we need two numbers that:
- Multiply to give
6 (the last number)
- Add to give
5 (the middle number)
Let’s find those numbers.
Possible pairs that multiply to 6:
- 1 and 6 → 1 + 6 = 7
✘
- 2 and 3 → 2 + 3 = 5
✔ ← This is it!
So the factors are
(x + 2)(x + 3)
But let’s show how each method works step by step.
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## Diamond Method:
In the diamond, you put:
- Top: product of a & c → here, a=1, c=6 → 1×6 =
6
- Bottom: sum of b → here, b=5 → so bottom is
5
Then you find two numbers that multiply to 6 and add to 5 → again,
2 and 3
Now, since the x² term has coefficient 1, we can directly write:
→ (x + 2)(x + 3)
✔ Done with Diamond Method.
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## XBox Method:
This uses a 2x2 box.
Step 1: Put x² in top-left, and 6 in bottom-right.
Step 2: We still need two numbers that multiply to 6 and add to 5 → again,
2 and 3
Put 2x and 3x in the other two boxes (order doesn’t matter).
So your box looks like:
| x² | 2x |
|------|------|
| 3x | 6 |
Step 3: Factor each row and column.
Top row: x² and 2x → common factor is
x → so first binomial starts with x
Bottom row: 3x and 6 → common factor is
3 → so second part is +3
Left column: x² and 3x → common factor is
x
Right column: 2x and 6 → common factor is
2
So reading across: (x + 2) and (x + 3)
Again, we get:
(x + 2)(x + 3)
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Both methods give the same answer. That’s good! It means we did it right.
Final Answer:
(x + 2)(x + 3)
Parent Tip: Review the logic above to help your child master the concept of diamond factoring worksheet.
