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Toolkit 10
Difference of squares
a
2
−
b
2
=
(
a
−
b
)
(
a
+
b
)
a^2-b^2=(a-b)(a+b)
a
2
−
b
2
=
(
a
−
b
)
(
a
+
b
)
Proof
Using the distributive law,
(
a
−
b
)
(
a
+
b
)
=
a
2
+
a
b
−
a
b
−
b
2
=
a
2
−
b
2
.
□
\begin{aligned} (a-b)(a+b) &= a^2 + ab - ab - b^2 \\ &= a^2 - b^2. \quad\square \end{aligned}
(
a
−
b
)
(
a
+
b
)
=
a
2
+
ab
−
ab
−
b
2
=
a
2
−
b
2
.
□