Thursday, 24 July 2008

Quadratic Equations | Special Factors

There are many mathematical methods to factor a quadratic equation.

But there is a group of quadratic equations that has a certain format that can be easily factorised without much computation.

They, therefore, deserve some attention!
They are the so-called "Special Factored Products" in this post.

Special Factored Products
  • x2 + 2xy + y2 = (x + y)2

  • x2 - 2xy + y2 = (x - y)2

  • x2 - y2 = ( x + y )( x - y )

NOTE: The centre term of the quadratic equation is the product of the square root of the two others and 2.

**They must ONLY be in these special form before we can proceed to apply the quick factorisation method**.

If you can be familiar with their quadratic equation format, you can translate them to their factorised form straight away without even computing !

Let's look at some example to enhance understanding of this important maths principles.

Example 1: Factorise x2 + 4x + 4
Applying the first factorisation format, the "y" can be replaced by 2 since y2 = 22.
Therefore the factored answer = (x + 2)2.

Example 2: Factorise 4a2 - 4a + 1
Applying the second factorisation format since now the maths sign for the term with power of 1 (that is the "a" term) is "minus", we can replace the "y" by 1 since 12 is 1.

How about the first term 4a2 which looks different now?
Using the knowledge of Indices, we know that 4a2 = 22a2 which becomes (2a)2.
We can use the understanding to replace the "x" term in the factorised form (x - y) 2.

Thus, the factorised form of 4a2 - 4a + 1 = (2a - 1)2.

Common mathematical mistakes made:

1) Forget to cater for the coefficient in the "x2" term. Take care of the number too!

2) The minus sign ONLY applies to the term with the power of 1 in the original unfactorised form. The term with pure number is still "+"!
If the quadratic equation has the minus sign in the number term, we cannot apply the second factorised form.

Example 3: Factorise 4a2 - 9
It is very obvious that this maths equation format is the third type.
Here we need to split the two terms into the "squared" format to convert the quadratic equation to the factorised product form.

Look at the first maths term. 4a2 = (2a)2 as in Example 2.
This is used to replace the "x" in the special factored product form (x - y)2.
The second term can be easily obtained to be 3 since 9 = 32 and the 3 will therefore replace the y term.

Quick factorising 4a2 - 9 ==> (2a - 3)(2a + 3). Simple?

It is good practice to remember this 3 special factored products and their conversion as I find that it saves time and many maths questions appear to be in this form.

Common problem:
Learners are not able to recognise this easy 2-term quadratic relation.
To be able to recognise this maths format, practice till the trick catches on ==> instinctive awareness!

Maths will be interesting if sufficient effort is put in. This goes to any thing in life too!

:-) :) :P


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