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Toolkit 6
Square of a sum
(
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
.
□