Quadratic Expressions Worksheets Page 11
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36number and algebra
• There are many combinations of numbers that satisfy m + n = b; however, only one
particular combination can be grouped and factorised. For example,
2
+
+ 12 = 2x
2
+
7x + 4x
+ 12 or 2x
2
+
+ 12 = 2x
2
+
8x + 3x
+ 12
2x
11x
11x
= 2x
+ 7x + 4x + 12
= 2x
+ 8x + 3x + 12
2
2
= x(2x + 7) + 4(x + 3)
= 2x(x + 4) + 3(x + 4)
= (x + 4) + (2x + 3)
cannot be factorised further
• In examining the general binomial expansion, a pattern emerges that can be used to help
identify which combination to use for m + n = b.
(dx + e) (fx + g) = dfx
+ dgx + efx + eg
2
= dfx
+ (dg + ef)x + eg
2
m + n = dg + ef and m × n = dg × ef
= b
= dgef
= dfeg
= ac
Therefore, m and n are factors of ac that sum to b.
• To factorise a general quadratic where a
1, look for factors of ac that sum to b. Then
rewrite the quadratic trinomial with four terms that can then be grouped and factorised.
+ bx + c = ax
+ mx + nx + c
2
2
ax
Factors of ac that sum to b
WOrKed eXamPle 6
WOrKed eXamPle 6
WOrKed eXamPle 6
− 11x − 10.
2
Factorise 6x
THInK
WrITe
− 11x − 10 = 6x
+ −11x + −10
2
2
6x
1
Write the expression and look for
common factors and special patterns.
The expression is a general quadratic
with a = 6, b = −11 and c = −10.
2
+ bx + c as
2
Since a
1, rewrite ax
Factors of −60
Sum of factors
+ mx + nx + c, where m and n are
2
ax
(6 × −10)
factors of ac (6 × −10) that sum to
− 60, 1
− 59
b (−11).
Calculate the sums of factor pairs of
− 20, 3
− 17
−60. As shown in blue, 4 and −15 are
− 30, 2
− 28
factors of −60 that add to −11.
15, − 4
11
−15, 4
−11
3
Rewrite the quadratic expression:
− 11x
− 10 = 6x
+
+
−15x
− 10
2
2
6x
4x
+ bx + c = ax
+ mx + nx + c
2
2
ax
with m = 4 and n = −15.
4
Factorise using the grouping
− 11x − 10 = 2x(3x + 2) + −5(3x + 2)
2
6x
2
+ 4x = 2x(3x + 2)
method: 6x
= (3x + 2) (2x − 5)
and −15x − 10 = −5 (3x + 2)
Write the answer.
Maths Quest 10 + 10A
278
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