J17

Similarly:
(a) (x + 3)(x + 4) = x(x + 4) + 3(x + 4)
= x 2 + 4x + 3x + 12
= x 2 + 7x + 12.
(b) (x + 2)(x + 7) = x(x + 7) + 2(x + 7)
= x 2 + 7x + 2x + 14
= x 2 + 9x + 14
(c) (x + a)(x + b) = x(x + b) + a(x + b)
= x 2 + bx + ax + ab
= x 2 + (a + b)x + ab
The last expression shows the relationship between the terms of the quadratic
expression and its factors. Thus, a constant term ab of a quadratic expression is the
product of the number terms a and b. whereas, the coefficient of x is the sum of a and b.
Example
Factorize x 2 + 10x + 21.
Solution
Compare x 2 + 10x + 21 and (x + __)(x + __), considering
x 2 + (a + b)x + ab = (x + a)(x + b).
This means x 2 + 10x + 21 = x 2 + (a + b)x + ab.
We need two numbers a and b such that a + b = 10 and ab = 21
Clearly, a and b are factors of 21 whose sum is 10. These are 3 and 7.
Rewriting x 2 + 10x + 21 as x 2 + (3 + 7)x + 21 gives x 2 + 3x + 7x + 21
= x(x + 3) + 7(x + 3)
= (x + 3)(x + 7).
Therefore, x 2 + 10x + 21 = (x + 3)(x + 7).
Example
Factorize x 2 + 7x + 12.
Solution
We need two numbers a and b such that a + b = 7 and ab = 12. These numbers are 3
and 4.
Thus, x 2 + 7x + 12 = x 2 + (3 + 4)x + 12
= x 2 + 3x + 4x + 12
= x(x + 3) + 4(x + 3)
= (x + 3)(x + 4)
Example
Factorize x2 – 6x + 8
Solution