Suppose, an algebraic expression in the form of \(ax^2+bx+c\) cannot be factorised directly using algebraic identities. We can factore it by splitting the middle term \((bx)\) into two terms whose product is \(ac\) and whose sum is \(b\).
How it works?
Consider the quadratic expression \(ax^2+bx+c\).
(1) Calculate \(ac\) (product of first and last term).
(2) Find two numbers whose product is ac and whose sum is \(b\).
(3) Split the middle term \(bx\) as those two numbers.
(4) Factorise by grouping.
(5) Take common factors from each group and write in factorised form.
(1) Calculate \(ac\) (product of first and last term).
(2) Find two numbers whose product is ac and whose sum is \(b\).
(3) Split the middle term \(bx\) as those two numbers.
(4) Factorise by grouping.
(5) Take common factors from each group and write in factorised form.
Example:
Factorise \(x^2+3x+2\).
Solution:
Given expression is \(x^2+3x+2\).
We need two numbers whose product is \(2\) and the sum is \(3\).
The only possible numbers are \(1\) and \(2\).
\(x^2+3x+2 =x^2+x+2x+2\)
\(=x(x+1)+2(x+1)\)
\(=(x+1)(x+2)\)
The factors are \((x+1)(x+2)\).
We need two numbers whose product is \(2\) and the sum is \(3\).
The only possible numbers are \(1\) and \(2\).
\(x^2+3x+2 =x^2+x+2x+2\)
\(=x(x+1)+2(x+1)\)
\(=(x+1)(x+2)\)
The factors are \((x+1)(x+2)\).