Special Products
Squares of binomials and the difference of squares follow patterns.
About this topic
Some polynomial products occur so often they are worth memorising. The square of a binomial is (a + b)² = a² + 2ab + b², and (a − b)² = a² − 2ab + b². The product of a sum and difference is (a + b)(a − b) = a² − b².
The middle term 2ab in a squared binomial is what students most often forget. For (x + 4)² the correct result is x² + 8x + 16, not x² + 16. The area model shows why: a square of side a + b contains a square a², a square b² and two rectangles of area ab. In the sum and difference pattern the middle terms cancel, so (x + 6)(x − 6) = x² − 36.
These patterns speed up calculation, even with numbers: 49 · 51 = (50 − 1)(50 + 1) = 2500 − 1 = 2499. They also lead directly to factoring perfect square trinomials and the difference of squares, and they appear in the derivation of the quadratic formula and the distance formula.
Check the signs. A squared difference has a negative middle term but a positive last term, since (−b)² is positive. Never write (a + b)² as a² + b², which loses the cross term.
Key ideas
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
Quick check
Use cases: where Special Products helps
- Homework: review the key idea behind Special Products before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Special Products without notes.
- Visual learning: play with the interactive lab to see how Special Products changes when you adjust its inputs.
- Teaching a friend: explain Special Products in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Special Products?
Squares of binomials and the difference of squares follow patterns.
What will I learn in Special Products?
(a + b)² = a² + 2ab + b². (a − b)² = a² − 2ab + b². (a + b)(a − b) = a² − b².
Is Special Products part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Special Products lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Special Products?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.