# Difference of Squares

Difference of squares (something squared minus something else squared):

$\color{purple}{a^2-b^2=\left(a-b\right)\left(a+b\right)}$

Proof of this fact is straightforward.

We just prove it from right to left.

Apply FOIL to ${\left({a}-{b}\right)}{\left({a}+{b}\right)}$:

${\left({a}-{b}\right)}{\left({a}+{b}\right)}={a}\cdot{a}+{a}\cdot{b}+{\left(-{b}\right)}\cdot{a}+{\left(-{b}\right)}\cdot{b}={{a}}^{{2}}+{a}{b}-{a}{b}-{{b}}^{{2}}={{a}}^{{2}}-{{b}}^{{2}}$.

This formula can also be derived by applying techniques from Factoring Quadratics note.

Note: formula is valid only for ${{a}}^{{2}}-{{b}}^{{2}}$. Sum of squares ${{a}}^{{2}}+{{b}}^{{2}}$ can't be factored at all.

Example 1. Factor ${{x}}^{{2}}-{9}$.

Notice, that ${9}={{3}}^{{2}}$.

Thus, ${{x}}^{{2}}-{9}={{x}}^{{2}}-{{3}}^{{2}}={\left({x}-{3}\right)}{\left({x}+{3}\right)}$.

Answer: ${{x}}^{{2}}-{9}={\left({x}-{3}\right)}{\left({x}+{3}\right)}$.

Of course, there can be more complex expressions.

Example 2. Factor ${9}{{y}}^{{2}}-{49}$.

Notice, that ${9}{{y}}^{{2}}={{\left({3}{y}\right)}}^{{2}}$ and ${49}={{7}}^{{2}}$.

Thus, ${9}{{y}}^{{2}}-{49}={{\left({3}{y}\right)}}^{{2}}-{{7}}^{{2}}={\left({3}{y}-{7}\right)}{\left({3}{y}+{7}\right)}$.

Answer: ${9}{{y}}^{{2}}-{49}={\left({3}{y}-{7}\right)}{\left({3}{y}+{7}\right)}$.

And even harder...

Example 3. Factor the following: ${{\left({x}+{y}\right)}}^{{2}}-{25}{{u}}^{{4}}{{b}}^{{6}}$.

Notice, that ${25}{{u}}^{{4}}{{b}}^{{6}}={{\left({5}{{u}}^{{2}}{{b}}^{{3}}\right)}}^{{2}}$.

Thus, ${{\left({x}+{y}\right)}}^{{2}}-{25}{{u}}^{{4}}{{b}}^{{6}}={{\left({x}+{y}\right)}}^{{2}}-{{\left({5}{{u}}^{{2}}{{b}}^{{3}}\right)}}^{{2}}={\left({\left({x}+{y}\right)}-{5}{{u}}^{{2}}{{b}}^{{3}}\right)}{\left({\left({x}+{y}\right)}+{5}{{u}}^{{2}}{{b}}^{{3}}\right)}$.

Answer: ${{\left({x}+{y}\right)}}^{{2}}-{25}{{u}}^{{4}}{{b}}^{{6}}={\left({x}+{y}-{5}{{u}}^{{2}}{{b}}^{{3}}\right)}{\left({x}+{y}+{5}{{u}}^{{2}}{{b}}^{{3}}\right)}$.

Now, it is time to exercise.

Exercise 1. Factor the following: ${{n}}^{{2}}-{36}$.

Answer: ${\left({n}-{6}\right)}{\left({n}+{6}\right)}$.

Exercise 2. Factor the following: $-{1}+{49}{{x}}^{{2}}$.

Answer: ${\left({7}{x}-{1}\right)}{\left({7}{x}+{1}\right)}$. Hint: $-{1}+{49}{{x}}^{{2}}={49}{{x}}^{{2}}-{1}$.

Exercise 3. Factor ${144}{{c}}^{{10}}{{d}}^{{8}}-{{\left({m}+{n}\right)}}^{{2}}$.

Answer: ${\left({12}{{c}}^{{5}}{{d}}^{{4}}-{m}-{n}\right)}{\left({12}{{c}}^{{5}}{{d}}^{{4}}+{m}+{n}\right)}$.

Exercise 4. Factor ${{\left({x}+{y}\right)}}^{{2}}-{{\left({x}-{y}\right)}}^{{2}}$.

Answer: ${4}{x}{y}$.