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Toolkit 7
Square of a difference
(
a
−
b
)
2
=
a
2
−
2
a
b
+
b
2
(a-b)^2=a^2-2ab+b^2
(
a
−
b
)
2
=
a
2
−
2
ab
+
b
2
Proof
Using the distributive law,
(
a
−
b
)
2
=
(
a
−
b
)
(
a
−
b
)
=
a
2
−
a
b
−
b
a
+
b
2
=
a
2
−
2
a
b
+
b
2
.
□
\begin{aligned} (a-b)^2 &= (a-b)(a-b) \\ &= a^2 - ab - ba + b^2 \\ &= a^2 - 2ab + b^2. \quad\square \end{aligned}
(
a
−
b
)
2
=
(
a
−
b
)
(
a
−
b
)
=
a
2
−
ab
−
ba
+
b
2
=
a
2
−
2
ab
+
b
2
.
□