Factor difference of squares expressions quickly

Expert intro: These are all difference-of-squares questions, so the pattern is the same every time: $A^2-B^2=(A-B)(A+B)$. Once you spot the square roots, the factoring is immediate.

Detailed walkthrough

Use the identity

$A^2-B^2=(A-B)(A+B)$

for each expression.

Answers

a) $x^2-49=(x-7)(x+7)$

b) $y^2-121=(y-11)(y+11)$

c) $4k^2-9=(2k-3)(2k+3)$

d) $16-144a^2=(4-12a)(4+12a)$

e) $9w^2-25x^2=(3w-5x)(3w+5x)$

f) $1-81p^2=(1-9p)(1+9p)$

💡 Pitfall guide

Watch the signs carefully. A lot of students try to force a single variable into both brackets, but the terms may contain different variables or a leading coefficient. First rewrite each term as a perfect square, then factor.

🔄 Real-world variant

If an expression is not a difference of squares but a sum of squares, this pattern will not work. For example, $x^2+49$ does not factor over the real numbers using this identity.

🔍 Related terms

difference of squares, factoring, algebraic identities