The quadratic expressions considered herein are of the form
ax2 + bx + c
where x is the (only) variable, and a, b and c are nonzero real numbers.
When a = 1, the quadratic expression simplifies to
x² + bx + c
If we set x² + bx + c equal to the product of x + c1 and x + c2, then
x² + bx + c = (x + c1) (x + c2)
= x² + (c1 + c2)x + c1c2
and thus b = c1 + c2 and c = c1c2.
Using this line of thinking, we can formulate a procedure for (possibly) factoring the quadratic expression x² + bx + c :
(1) Find two real numbers whose sum is equal to b and whose product is equal to c.
(2) If two real numbers satisfying the above criteria are found, and we denote them as c1 and c2 , then (x + c1) and (x + c2) are factors of x² + bx + c.
In most cases, b and c are integers, so c1 and c2 will also be integers.
Obviously, there are quadratic expressions of the type x²+ bx + c for which no two real numbers can be found, whose sum is equal to b and whose product is equal to c. Either they don't exist, or they are not easily determined (especially when they are not integers). In this case, a different procedure for factoring is called for.
EX.
Factor x² + 7x + 12
First, list the combinations of integers whose product is equal to 12.
They are
12 and 1 ( 12 · 1 = 12 )
6 and 2 ( 6 · 2 = 12 )
4 and 3 ( 4 · 3 = 12 )
Then find the sum of these combinations:
12 + 1 = 13
6 + 2 = 8
4 + 3 = 7
The integers 4 and 3 have a sum of 7 and a product of 12.
\ The factors of x²+ 7x + 12 are ( x + 3 ) and ( x + 4 ).
EX.
Factor y²- 3y - 10
The combinations of integers whose product is -10, and their sums are
10 · -1 = -10 10 + ( -1 ) = 9
5 · -2 = -10 5 + ( - 2) = 3
-10 · 1 = -10 -10 + 1 = -9
-5 · 2 = -10 -5 + 2 = -3
The integers -5 and 2 have a sum of -3 and a product of -10.
\ The factors of y²- 3y - 10 are ( y -5 ) and ( y + 2 ).
The quadratic expression ax²+ bx + c can be rewritten as
If b/a and c/a are integers, then the quadratic expression
can be factored using the above procedure. Oftentimes, though, this is not the case. Thus, a different procedure is utilized to factor this quadratic expression.
The quadratic expression ax²+ bx + c can be algebraically manipulated as follows:
ax²+ bx + c = ax²+ (b1 + b2)x + c
= ax²+ b1 x + b2x + c
If b1 b2 = ac, then (b1 b2 / a) = c, and thus
Therefore, the following procedure can be used to factor the expression
ax²+ bx + c:
(1) Find two real numbers b1 and b2 whose sum is b and whose product is equal to the product of a and c.
(2) If two real numbers b1 and b2 are found satisfying the above criteria, then
are factors of the expression ax²+ bx + c and may be further simplified.
This procedure is most useful when a, b and c are integers. Of course, this method will not always work, since there is no guarantee that the numbers b12 can be found. Other factoring techniques will have to be employed in these cases. and b
EX.
Factor 6x²+ 19x + 10.
As a = 6, b = 19 and c = 10, ac = (6)(10) = 60. Some combinations of integers whose product is 60, along with their sums, are
60 · 1 = 60, 60 + 1 = 61
30 · 2 = 60, 30 + 2 = 32
20 · 3 = 60, 20 + 3 = 23
15 · 4 = 60, 15 + 4 = 19
The two numbers 15 and 4 have a sum of 19 and a product of 60.
Letting b1 = 15 and b2 = 4, we can express 6x² + 19x + 10 as the product of
Therefore, 6x² + 19x + 10
= (1.5x + 1) (4x + 10)
= (1.5x + 1) 2 (2x + 5)
= (3x + 2) (2x + 5)
EX.
Factor 10x+ 7x - 12.
As a = 10, b = 7 and c = -12, ac = (10)(-12) = -120.
Some combinations of integers whose product is 120, along with their sums, are
-10 · 12 = -120, -10 + 12 = 2
10 · -12 = -120, 10 + ( -12) = -2
-15 · 8 = -120, -15 + 8 = -7
15 · -8 = -120, 15 + ( -8) = 7
The two numbers 15 and -8 have a sum of 7 and a product of 120. Letting b1 = 15 and b2 = -8, we can express 10x²+ 7x - 12 as the product of
Therefore, 10x+ 7x - 12 = (-1.25x + 1) (-8x - 12)
= (-1.25x + 1) -4 (2x + 3)
= (5x - 4) (2x + 3)
