Same Math › Algebra 1 · grades 8–10 › The X method
What is the X method for factoring?
The X method is a little puzzle. Draw an X, put the product on top and the sum on the bottom, and find the two side numbers that multiply to the top and add to the bottom. For x² + 7x + 12, the top is 12 and the bottom is 7, so the sides are 3 and 4, and the answer is (x + 3)(x + 4).
- Top: the product the two numbers make (the last term, or a × c).
- Bottom: the sum they make (the middle number).
- Sides: the two numbers. They become the numbers in the factors.
Why schools teach it
It isolates the one hard step
Every way of factoring x² + bx + c comes down to one question: which two numbers multiply to c and add to b? (HSA.SSE.B.3a). The X pulls that question out and drills it, often weeks before factoring, as diamond puzzles in pre-algebra.
When there's a number in front of x², the top of the X becomes a × c. The two side numbers then split the middle term for the box or for grouping.
At the kitchen table
What to say, and what to skip
Say this
What multiplies to the top number?
List the factor pairs.Which pair adds to the bottom number?
If the top is negative, what do you know about the signs?
One positive, one negative.Now where do those two numbers go?
Into the factors, or into the box.
Skip this
- Only trying positive factor pairs.−3 and 5 multiply to −15 and add to 2.
- Putting b on top and c on the bottom.Product on top, sum on the bottom.
Just guess.
Listing factor pairs in order finds it every time.
See it in action
Factoring x² + 2x − 24
Top: −24. Bottom: 2
List pairs for −24 until one adds to 2
−4 × 6 = −24 and −4 + 6 = 2Use the sides as the numbers in the factors
(x − 4)(x + 6)
With a number in front, like 2x², the top is a × c and the sides split the middle term.
Free practice sheet
Make an X method sheet
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Parents also ask
Why is it called the diamond method?
Some teachers draw a diamond instead of an X, with the same four spots. It's the same puzzle.
What goes on top when the x² has a number, like 2x²?
a × c. For 2x² + 7x + 3, the top is 6 and the bottom is 7, so the sides are 1 and 6. Then use the box or grouping to finish.
What if the top is 0?
Then there's no constant term, like x² + 5x. Just pull out the GCF: x(x + 5).
How do kids find the pair faster?
List factor pairs of the top in order: 1 and 24, 2 and 12, 3 and 8, 4 and 6. Then try the signs.