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**

- x
^{2}+ 2xy + y^{2}= (x + y)^{2} - x
^{2}- 2xy + y^{2}= (x - y)^{2} - x
^{2}- y^{2}= ( 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 x

^{2}+ 4x + 4

Applying the first factorisation format, the "y" can be replaced by 2 since y

^{2}= 2

^{2}.

Therefore the factored answer = (x + 2)

^{2}.

__Example 2__: Factorise 4a

^{2}- 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 1

^{2}is 1.

*How about the first term 4a*

^{2}which looks different now?Using the knowledge of Indices, we know that 4a

^{2}= 2

^{2}a

^{2}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 4a

^{2}- 4a + 1 = (2a - 1)

^{2}.

**Common mathematical mistakes made**:

1) Forget to cater for the coefficient in the "x

^{2}" 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 4a

^{2}- 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. 4a

^{2}= (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 = 3

^{2}and the 3 will therefore replace the y term.

Quick factorising 4a

^{2}- 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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