Same Math › Algebra 1 · grades 8–10 › Difference of squares and perfect squares
What are the special factoring patterns?
Two shapes factor on sight. A difference of squares, like x² − 9, is (x + 3)(x − 3). A perfect square trinomial, like x² + 6x + 9, is (x + 3)², because the first and last terms are squares and the middle is twice their product. Kids learn to spot the pattern instead of working it out each time.
- a² − b²: two squares with a minus between them.
- a² ± 2ab + b²: first and last are squares, middle is 2 × a × b.
- A sum of squares, like x² + 9, doesn't factor with real numbers.
Why schools teach it
Seeing structure is the skill
The standard asks kids to see structure, like noticing that x⁴ − y⁴ is (x²)² − (y²)² (HSA.SSE.A.2). These two patterns are the main examples.
They also explain shortcuts used all year: completing the square builds a perfect square on purpose, and the difference of squares makes fast mental math like 21 × 19 = 400 − 1.
At the kitchen table
What to say, and what to skip
Say this
Is the first term a perfect square? The last?
Is there a minus sign between two squares?
Difference of squares.Is the middle term twice the product of the square roots?
Perfect square.Can you check it by multiplying back?
Skip this
x² + 9 is (x + 3)(x + 3).
That's x² + 6x + 9. A sum of squares doesn't factor.- Writing (x − 3)² for x² − 9.(x − 3)² has a middle term: x² − 6x + 9.
- Forgetting the GCF.2x² − 50 = 2(x² − 25) = 2(x + 5)(x − 5).
See it in action
Factoring 4x² − 12x + 9
Check the ends
4x² = (2x)² and 9 = 3²Check the middle: 2 × 2x × 3
12x, with a minus signIt's a perfect square
(2x − 3)²
If the middle didn't match, the box or X method still works.
Free practice sheet
Make a factoring patterns sheet
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Parents also ask
Why doesn't x² + 9 factor?
Two numbers that multiply to 9 and add to 0 would have to be 3 and −3, but they multiply to −9. So no real factors exist.
Do these show up on tests?
Yes, often as quick factoring and in solving equations like x² − 49 = 0.
How is the difference of squares used in mental math?
21 × 19 = (20 + 1)(20 − 1) = 400 − 1 = 399.
What's a perfect square trinomial for?
Completing the square makes one on purpose, which leads to the quadratic formula.