Factoring an expression means representing the expression as multiple of two or more terms. In a quadratic equation, that is an equation in which one of the term contains square of the variable (say x), and no term which has higher power than square. There are always maximum two factors containing the term x. Thus the factors of the expression of the type:
ax^2 + bx + c
will be of the form:
(x - A)(x -B)*C
Where A, B and C are constant numbers.
There are formulae to calculate the values of A, B and C from the values of a, b and c, without finding out the factors. But in many cases it is possible to find out the factors with a little bit of ingenuity or a little bit of trial and error. To illustrate how this can be done we will find out factors of the given expression in two different ways.
Method I:
If we examine the expression 9x^2 - 6x + 1 carefully, we can see that it conforms to the form p^2 - 2pq - q^^2, which is equal to (p - q)^2. Thus we can straight away we can find out the factors of the expression as:
(3x - 1)^2 = (x - 1/3)(x -1/3)*9
Method II
In this method we need term bx of the expression ax^2 + bx + c in two components b'x + b"x so that we can find a common factor in the terms (ax^ + b'x) and (b"x + c).
Thus we can represent the given expression as:
9x^2 - 3x - 3x + 1
= 3x*(3x - 1) -1*(3x - 1)
= (3x - 1)(3x - 1)
= (x - 1/3)(x - 1/3)*9
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