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Toolkit 19
Factoring quadratic by grouping
x
2
+
(
a
+
b
)
x
+
a
b
=
(
x
+
a
)
(
x
+
b
)
x^2+(a+b)x+ab=(x+a)(x+b)
x
2
+
(
a
+
b
)
x
+
ab
=
(
x
+
a
)
(
x
+
b
)
Proof
Using the distributive law,
(
x
+
a
)
(
x
+
b
)
=
x
2
+
b
x
+
a
x
+
a
b
=
x
2
+
(
a
+
b
)
x
+
a
b
.
□
\begin{aligned} (x+a)(x+b) &= x^2+bx+ax+ab \\ &= x^2+(a+b)x+ab. \quad\square \end{aligned}
(
x
+
a
)
(
x
+
b
)
=
x
2
+
b
x
+
a
x
+
ab
=
x
2
+
(
a
+
b
)
x
+
ab
.
□