Quadratic Expressions
In the package on Factorizing Expressions we looked at how to
factorize quadratic expressions which have the number 1 in front
of the highest order term, x,
y, z, etc.. If the highest order term has a
number other than this then more work must be done to factorize
the expression. As in the earlier case, some insight is gained by
looking at a general expression with factors (ax + c) and (bx +
d). Then
showing that the coeffcient of the square term, x,
is ab, the product of the coeffcients of the xterms in each
factor. The coeffcient of the xterm is made up from the
coeffcients as follows:
This is the information needed to find the factors of
quadratic expressions.
Example 1
Factorize the following expressions.
(a) 2x
+ 7x + 3 , (b) 10x
+ 9x + 2 .
Solution
(a) The factors of 2 are 2 and 1, and the
factors of 3 are 3 and 1. If the quadratic expression factorizes
then it is likely to be of the form (2x + c)(1x + d) and the
choice for c, d is 3, 1 or 1, 3. Trying the first combination,
(2x + 3)(x + 1) = 2x
+ 2x + 3x + 3 ,
= 2x + 5x + 3 (which is incorrect) .
The second choice is
(2x + 1)(x + 3) = 2x+
6x + x + 3 ,
= 2x+ 7x + 3 , which is therefore the correct
factorization.
(b) There is more than one choice for the
first term since 10 is 1 Ã— 10 as well as 2 Ã— 5. The final term
will factor as 2 Ã— 1. Which combination of pairs, either (1, 10)
with (2, 1), or (2, 5) with (2, 1), will give the correct
coeffcient of x, i.e., 9? The latter two pairs seem the more
likely since 2 Ã— 2 + 5 Ã— 1 = 9. Checking
(2x + 1)(5x + 2) = 10x
+ 4x + 5x + 2 ,
= 10x + 9x + 2 .
Exercise 1.
Factorize each of the following expressions.
(a) 2x + 5x + 3
(b) 3x + 7x + 2
(c) 3y  5y  2
(d) 4z  23z + 15
(e) 64z + 4z  3
(f) 4w  25
Solution
(a) In this case we have 2x
+ 5x + 3 = (2x + 3)(x + 1)
(b) In this case we have 3x
+ 7x + 2 = (3x + 1)(x + 2)
(c) In this case we have 3y
 5y  2 = (3y + 1)(y  2)
(d) In this case we have 4z
 23z + 15 = (4z  3)(z  5)
(e) In this case we have 64z
+ 4z  3 = (16z  3)(4z + 1)
(f) This is a case of the difference of two squares which was
seen in the package on Brackets. 4w
 25 = (2w  5)(2w + 5)
Quiz
To which of the following does 12x2 + 17x  14 factorize?
(a) (12x + 7)(x  2) (b) (x + 2)(12x  7) (c) (4x + 7)(x  3)
(d) (x  7)(4x + 3)
Solution
There are several possibilities since the final term is 14
and the two quantities corresponding to c and d must therefore
have opposite signs. The possible factors of 12 are (1, 12), (2,
6), (3, 4). For 14, the possible factors are (Â±1,14), (Â±2,7). It is now a matter of trial and error. The
possible combinations are
(1, 12) and (Â±1,14) , (1, 12) and (Â±2,7) ,
(2, 6) and (Â±1,14) , (2, 6) and (Â±2,7) ,
(3, 4) and (Â±1,14) , (3, 4) and (Â±2,7) .
By inspection (2 Ã— 12) + (1 Ã— {7}) = 24  7 = 17, so the
factors appear to be (x + 2) and (12x  7). This can easily be
checked.
(x + 2)(12x  7) = 12x
 7x + 24x  14 ,
= 12x+ 17x  14 ,
and the required factorization has been achieved.
