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Difference of Squares

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

a2−b2=(a−b)(a+b)\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 (a−b)(a+b){\left({a}-{b}\right)}{\left({a}+{b}\right)}:

(a−b)(a+b)=a⋅a+a⋅b+(−b)⋅a+(−b)⋅b=a2+ab−ab−b2=a2−b2{\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 a2−b2{{a}}^{{2}}-{{b}}^{{2}}. Sum of squares a2+b2{{a}}^{{2}}+{{b}}^{{2}} can't be factored at all.

Example 1. Factor x2−9{{x}}^{{2}}-{9}.

Notice, that 9=32{9}={{3}}^{{2}}.

Thus, x2−9=x2−32=(x−3)(x+3){{x}}^{{2}}-{9}={{x}}^{{2}}-{{3}}^{{2}}={\left({x}-{3}\right)}{\left({x}+{3}\right)}.

Answer: x2−9=(x−3)(x+3){{x}}^{{2}}-{9}={\left({x}-{3}\right)}{\left({x}+{3}\right)}.

Of course, there can be more complex expressions.

Example 2. Factor 9y2−49{9}{{y}}^{{2}}-{49}.

Notice, that 9y2=(3y)2{9}{{y}}^{{2}}={{\left({3}{y}\right)}}^{{2}} and 49=72{49}={{7}}^{{2}}.

Thus, 9y2−49=(3y)2−72=(3y−7)(3y+7){9}{{y}}^{{2}}-{49}={{\left({3}{y}\right)}}^{{2}}-{{7}}^{{2}}={\left({3}{y}-{7}\right)}{\left({3}{y}+{7}\right)}.

Answer: 9y2−49=(3y−7)(3y+7){9}{{y}}^{{2}}-{49}={\left({3}{y}-{7}\right)}{\left({3}{y}+{7}\right)}.

And even harder...

Example 3. Factor the following: (x+y)2−25u4b6{{\left({x}+{y}\right)}}^{{2}}-{25}{{u}}^{{4}}{{b}}^{{6}}.

Notice, that 25u4b6=(5u2b3)2{25}{{u}}^{{4}}{{b}}^{{6}}={{\left({5}{{u}}^{{2}}{{b}}^{{3}}\right)}}^{{2}}.

Thus, (x+y)2−25u4b6=(x+y)2−(5u2b3)2=((x+y)−5u2b3)((x+y)+5u2b3){{\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: (x+y)2−25u4b6=(x+y−5u2b3)(x+y+5u2b3){{\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: n2−36{{n}}^{{2}}-{36}.

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

Exercise 2. Factor the following: −1+49x2-{1}+{49}{{x}}^{{2}}.

Answer: (7x−1)(7x+1){\left({7}{x}-{1}\right)}{\left({7}{x}+{1}\right)}. Hint: −1+49x2=49x2−1-{1}+{49}{{x}}^{{2}}={49}{{x}}^{{2}}-{1}.

Exercise 3. Factor 144c10d8−(m+n)2{144}{{c}}^{{10}}{{d}}^{{8}}-{{\left({m}+{n}\right)}}^{{2}}.

Answer: (12c5d4−m−n)(12c5d4+m+n){\left({12}{{c}}^{{5}}{{d}}^{{4}}-{m}-{n}\right)}{\left({12}{{c}}^{{5}}{{d}}^{{4}}+{m}+{n}\right)}.

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

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