Factoring





Factoring-Quadratic equations:

An equation which is in the form of ax² + b x + c =0,where a,b,c ∈ R and a≠0 is called a quadratic equation. Finding the roots of a quadratic equation is called as solving the quadratic equation.

Procedure:

 (1) Arrange the terms of the expression in the standard form.

 (2) Find the product of coefficients of x² and the constant term.

 (3) Split that value into two terms and the simplified value must be equal to the middle term.

 (4) Therefore  x² + (a+b) x + ab = (x+a) (x+b)

Equation

Symbol of two terms

ax² + b x + c = 0

The symbol of two terms must be positive.

ax² - b x + c = 0

The symbol of two terms must be negative.

ax² + b x - c = 0

The small number of those will have negative sign.

ax² - b x - c = 0

The large number of those will have negative sign.

Example 1:

 x² + 17 x + 60

Solution:

We don't have any number before square term. So that we can take the last term 60. Then we have to split 60 as two numbers and the simplified value two numbers must be equal to the middle term.

                    = x² + 17 x + 60

                    = x² + 12 x + 5 x  + 60

                    = x(x + 12) + 5 (x  + 12)

                    = (x + 12) (x  + 5)


Example 2

 x² - 5 x - 36

Solution:

We don't have any number before square term. So that we can take the last term -36. Then we have to split -36 as two numbers and the simplified value two numbers must be equal to the middle term.

                    = x² - 5 x - 36

                    = x² - 9 x + 4x - 36

                    = x(x - 9) + 4 (x - 9)

                    = (x + 4) (x  - 9)


Example 3

Factorize x² - 14 x + 48

Solution:

We don't have any number before square term. So that we can take the last term 48. Then we have to split 48 as two numbers and the simplified value two numbers must be equal to the middle term.

                    = x² - 14 x + 48

                    = x² - 6 x - 8x + 48

                    = x(x - 6) - 8 (x - 6)

                    = (x - 6) (x  - 8)


Example 4

Factorize x² + 2 x - 24

Solution:

We don't have any number before square term. So that we can take the last term -24. Then we have to split -24 as two numbers and the simplified value two numbers must be equal to the middle term.

                    = x² + 2 x - 24

                    = x² + 6 x - 4 x -24

                    = x(x + 6) - 4 (x + 6)

                    = (x + 6) (x  - 4)