Difference of Squares
a² − b² factors as (a + b)(a − b).
About this topic
A difference of squares has the form a² − b² and factors as (a + b)(a − b). For example x² − 49 = (x + 7)(x − 7). Both terms must be perfect squares and subtracted.
A sum of squares like x² + 49 does not factor over the real numbers. Sometimes you can apply the pattern more than once: x⁴ − 16 = (x² + 4)(x² − 4) = (x² + 4)(x + 2)(x − 2). Coefficients can be squares too, so 25x² − 9 = (5x + 3)(5x − 3). The reason it works is that when you multiply (a + b)(a − b) the middle terms are opposites and cancel.
The pattern gives mental math shortcuts, such as 101 · 99 = 100² − 1 = 9999. It is used to rationalize denominators containing radicals, since multiplying by a conjugate removes the root, and in geometry when finding the area of a ring, which is the difference of two circle areas.
Check for a common factor first: 2x² − 18 = 2(x² − 9) = 2(x + 3)(x − 3). Do not try to factor a sum, and do not forget that every term must be a perfect square, so x² − 10 does not fit the pattern with whole numbers.
Key ideas
- Both terms must be perfect squares
- Subtraction between them
- Sum of squares does not factor over the reals
Quick check
Use cases: where Difference of Squares helps
- Homework: review the key idea behind Difference of Squares before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Difference of Squares without notes.
- Visual learning: play with the interactive lab to see how Difference of Squares changes when you adjust its inputs.
- Teaching a friend: explain Difference of Squares 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 Difference of Squares?
a² − b² factors as (a + b)(a − b).
What will I learn in Difference of Squares?
Both terms must be perfect squares. Subtraction between them. Sum of squares does not factor over the reals.
Is Difference of Squares 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 Difference of Squares 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 Difference of Squares?
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.