#### Topic Content:

- Factorization of Perfect Squares

Expressions such as (x + y)^{2} and (x – y)^{2 }are examples of perfect squares.

If we expand (x + y)^{2}, we obtain (x + y)^{2} = (x + y)(x + y)

= x^{2 }+ xy + xy + y^{2}

= x^{2} + 2xy + y^{2}

The result of this equation may be stated as follows:

(1^{st} term)^{2} + twice the product of the first term and the 2^{nd} term + (2^{nd} term)^{2}

Similarly, if we expand (x – y)^{2}, we obtain

(x – y)^{2} = (x – y)(x + y)

=x^{2} – xy – xy + y^{2}

=x^{2} – 2xy + y^{2}

The result is also:

(1^{st} term)^{2} – twice the product of the first term and the 2^{nd} term + (2^{nd} term)^{2}

It is important to know that expressions of the form x^{2} + 2xy + y^{2} and x^{2} – 2xy + y^{2} are also known as perfect squares.

### Example 6.4.1:

Factorise the following:**(a)** \( \scriptsize 16 \: – \: 8x \: + \: x^2 \)**(b)** \( \scriptsize 4x^2 \: + \: 36x \: + \: 81 \)**(c)** \( \scriptsize 1 \: – \: \normalsize \frac {2x}{3} \: + \: \frac {x^2}{9} \)**(d) **\( \scriptsize 16x^2 \: – \: 64xy \: + \: 64y^2 \)

**Solution**

**(a)** \( \scriptsize 16 \: – \: 8x \: + \: x^2 \)

Rewrite as;

\( \scriptsize x^2 \: – \: 8x \: + \: 16 \)

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