(a + b)(a - b) is the difference of two squares pattern in Intermediate Algebra. It multiplies two conjugates to give a^2 - b^2, which makes factoring and simplifying faster.
(a + b)(a - b) is the special product that gives you a difference of squares: (a + b)(a - b) = a^2 - b^2. In Intermediate Algebra, you use it when you see two binomials with the same two terms, but one has a plus and the other has a minus.
The reason this works is the cross terms cancel. If you multiply with FOIL, you get a^2 - ab + ab - b^2. The middle terms are opposites, so they disappear, leaving just a^2 - b^2. That cancellation is what makes this pattern so useful and so easy to spot once you know what to look for.
This is not just a memorized formula. It is the factoring form of a difference of squares. If you start with a^2 - b^2, you can factor it into (a + b)(a - b). If you start with the product (a + b)(a - b), you expand it back into the difference of squares. Both directions matter, because problems can ask you to factor, expand, simplify, or solve.
A common Intermediate Algebra example is x^2 - 49. Since 49 is 7^2, you can rewrite it as x^2 - 7^2 and factor it as (x + 7)(x - 7). The same move works with variables, numbers, and expressions, as long as each term is a perfect square. For example, 9m^2 - 16 becomes (3m + 4)(3m - 4).
The biggest check is that you have a subtraction between two squares, not a sum. If you see a plus sign, this pattern does not apply. Also, both pieces need to be perfect squares before you factor. If one term is not a square, you may need a different factoring method or a different algebra step first.
This pattern shows up all over Intermediate Algebra because it turns a hard-looking multiplication or factoring problem into a quick recognition task. Once you can spot a difference of squares, you can simplify expressions faster instead of using trial and error or multiplying everything out from scratch.
It also shows up when you solve equations. If you get something like x^2 - 25 = 0, factoring gives (x + 5)(x - 5) = 0, and then you can use the zero product property. That connection between special products and solving equations is a big reason this term keeps coming back in the course.
You will also see it in rational expressions and simplification. Expressions like (x^2 - 16)/(x - 4) can often be factored first so you can reduce common factors. That makes the algebra cleaner and helps you avoid getting stuck with a giant expression that looks harder than it really is.
This is one of the first places where pattern recognition starts to matter more than brute-force expanding. If you can tell when a problem fits the difference of squares structure, you save time and cut down on sign mistakes.
Keep studying Intermediate Algebra Unit 6
Visual cheatsheet
view galleryDifference of Two Squares
This is the pattern behind (a + b)(a - b). The product expands to a^2 - b^2, and the reverse form lets you factor a difference of two squares whenever both terms are perfect squares. If you can recognize the square roots quickly, factoring becomes almost automatic.
FOIL Method
FOIL is the expansion move that shows why the pattern works. When you multiply the binomials, the outer and inner terms cancel because they are opposites. That cancellation is what turns the product into a^2 - b^2, so FOIL is a good way to check your answer.
Factoring
This pattern is one specific factoring strategy inside the broader skill of factoring polynomials. Instead of testing random binomial factors, you recognize a special form right away. That saves time and is especially useful when the expression is already set up as a subtraction of two squares.
Perfect Square Trinomial
This is a nearby factoring pattern, but it looks different. A perfect square trinomial has three terms and comes from squaring a binomial, while (a + b)(a - b) has two binomials that multiply to a difference of squares. Students often mix them up because both involve square roots and binomials.
A quiz or problem set will usually ask you to factor an expression like x^2 - 64, simplify a product, or solve a quadratic that turns into a difference of squares. Your job is to check whether both terms are perfect squares, then rewrite the expression in the form (a + b)(a - b). If the problem is solving an equation, you usually factor first and then set each factor equal to zero.
You may also need to show the multiplication step if the teacher wants expansion. In that case, use FOIL or distributive property to prove that the middle terms cancel. On word problems or mixed review sets, this pattern often appears hidden inside a larger expression, so careful rewriting matters as much as the final answer.
These two get mixed up because they both involve squares and factoring. A perfect square trinomial has three terms, like a^2 + 2ab + b^2, and it factors into (a + b)^2. The expression (a + b)(a - b) is different because it has two binomials with opposite middle signs and expands to a^2 - b^2.
(a + b)(a - b) is the difference of two squares pattern, and it expands to a^2 - b^2.
The middle terms cancel when you multiply, which is why this special product is faster than regular FOIL work.
You can factor a difference of squares only when both terms are perfect squares and the expression is subtraction, not addition.
This pattern is useful for simplifying expressions, factoring quadratics, and solving equations like x^2 - 25 = 0.
If you see a pair like (x + 7)(x - 7), you should think x^2 - 49 immediately.
It is a special product called the difference of squares pattern. When you multiply the two binomials, the result is a^2 - b^2. In Intermediate Algebra, you use this pattern both to expand expressions and to factor expressions that already look like a difference of two squares.
Rewrite each term as a square, then take the square root of both pieces and make one binomial with a plus sign and one with a minus sign. For example, x^2 - 36 becomes (x + 6)(x - 6). The key check is that the expression must be subtraction, not addition.
They cancel because one is +ab and the other is -ab. If you expand with FOIL, you get a^2 + ab - ab - b^2. Since +ab and -ab are opposites, they add to zero.
Yes. If you have an equation like x^2 - 81 = 0, you can factor it as (x + 9)(x - 9) = 0 and then set each factor equal to zero. That turns one quadratic equation into two simpler linear equations.