Key takeaway: A difference of squares is a binomial of the form a² − b², one perfect square minus another, and it always factors as (a + b)(a − b).
Grade 9 – Algebra 1A-SSE.A.2Expressions and Equations
Definition of Difference of Squares
A difference of squares is a two-term expression in which both terms are perfect squares separated by a minus sign, such as \(x^{2} – 81\) or \(49x^{2} – 144\). It has a pattern that factors immediately, with no guessing: one factor has a plus, the other has a minus.
Why does it work? Multiply \((a + b)(a – b)\) and the middle terms \(-ab\) and \(+ab\) cancel, leaving \(a^{2} – b^{2}\). A picture helps: cut a small square out of a bigger square, and the leftover L-shape can be rearranged into a rectangle with sides \(a + b\) and \(a – b\). The pattern also speeds up mental math, such as \(97 \times 103 = 100^{2} – 3^{2} = 9991\).
Formula and notation
The difference of squares identity, read in either direction, is
\[a^{2} – b^{2} = (a + b)(a – b)\]
To use it, take the square root of each term to find \(a\) and \(b\). For \(49x^{2} – 144\), \(a = 7x\) and \(b = 12\).
Key facts
- A difference of squares needs two perfect squares with a minus sign between them.
- The two factors are conjugates: same terms, opposite middle signs.
- A sum of squares such as x² + 25 cannot be factored over the real numbers.
- Always pull out a greatest common factor first, then look for the pattern.
Difference of Squares example
Factor \(49x^{2} – 144\).
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Check for squares: \(49x^{2} = (7x)^{2}\) and \(144 = 12^{2}\), separated by a minus.
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Set \(a = 7x\) and \(b = 12\).
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Apply the pattern: \((7x + 12)(7x – 12)\). Check: \(49x^{2} – 84x + 84x – 144 = 49x^{2} – 144\).
\(49x^{2} – 144 = \) \((7x + 12)(7x – 12)\).
Common mistake
Watch out
Zane factors \(x^{2} + 25\) as \((x + 5)(x – 5)\).
Why it is wrong
The pattern needs a minus sign. Multiplying \((x + 5)(x – 5)\) gives \(x^{2} – 25\), not \(x^{2} + 25\). A sum of two squares has no factorization with real numbers, so \(x^{2} + 25\) is prime.
Related terms and lessons
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