THE FACTORISATION OF 9x -
18x is
Answers
The quadratic:
9
x
2
+
18
x
+
8
is in the form:
a
x
2
+
b
x
+
c
with
a
=
9
,
b
=
18
and
c
=
8
.
This has discriminant
Δ
given by the formula:
Δ
=
b
2
−
4
a
c
=
18
2
−
4
(
9
)
(
8
)
=
324
−
288
=
36
=
6
2
Since this is positive and a perfect square, this quadratic will factor with integer coefficients.
Let's use an AC method:
Find a pair of factors of
A
C
=
9
⋅
8
=
72
with sum
B
=
18
. (We look for a sum rather than a difference since the sign of
c
is positive).
The pair
12
,
6
works in that
12
⋅
6
=
72
and
12
+
6
=
18
.
Use this pair to split the middle term and factor by grouping:
9
x
2
+
18
x
+
8
=
(
9
x
2
+
12
x
)
+
(
6
x
+
8
)
9
x
2
+
18
x
+
8
=
3
x
(
3
x
+
4
)
+
2
(
3
x
+
4
)
9
x
2
+
18
x
+
8
=
(
3
x
+
2
)
(
3
x
+
4
)
We can find the same factorisation by completing the square:
Note that
9
x
2
=
(
3
x
)
2
and
18
x
=
2
(
3
x
)
(
3
)
.
Hence we find:
9
x
2
+
18
x
+
8
=
(
3
x
)
2
+
2
(
3
x
)
(
3
)
+
3
2
−
1
9
x
2
+
18
x
+
8
=
(
3
x
+
3
)
2
−
1
2
9
x
2
+
18
x
+
8
=
(
(
3
x
+
3
)
−
1
)
(
(
3
x
+
3
)
+
1
)
9
x
2
+
18
x
+
8
=
(
3
x
+
2
)
(
3
x
+
4
)