DIFFERENCE OF TWO SQUARES

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The difference of the squares of two quantities is the product of its sum and their difference.

This means the difference of the square of two quantities(say $x$ and $y$) is:
 $x^2-y^2=(x+y)(x-y)$

You might be asking "how true is this?". Look at this!!!

Let expand the bracket
$x(x-y)+y(x-y)$
$x^2-xy+xy-y^2$
$x^2-y^2$.....  which is true.

Let practice with the following examples

Example 1
Factorizes $x^2-16$

$(x)^2-(4)^2$
By the difference of two squares
$(x+4)(x-4)$

Example 2
Factorize $64-49x^2$

$(8)^2-(7x)^2$
By the difference of two squares
$(8+7x)(8-7x)$

Example 3
Factorize the expression $a^2b^2-9c^2$

$(ab)^2-(3c)^2$
By difference of two squares
$(ab+3c)(ab-3c)$

In many cases, you may be required to factories before taking the difference of two squares.

Example 4
Factorize the expression $240x^2-60$

$60$ is common in both sides
$60(4x^2-1)$
$60((2x)^2-(1)^2)$

By taking the difference of two squares
$60((2x+1)(2x-1))$

Example 5
Factorize the expression $250x^2-160y^2$

$10$ is common in both sides
$10(25x^2-16y^2)$
$10((5x)^2-(4y)^2)$

By taking the difference of two squares
$10(5x+4y)(5x-4y))$

The difference of two squares can be used to simplify the calculation, as is the case of the next example.

Example 6
Calculate $44^2-20^2$

$44^2-20^2=(44+20)(44-20)$
                      $=(64)(24)$
                      $=1536$

Check:
$44^2=1936$, and $20^2=400$
$1936-400=1536$ which correspond with our answer.

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Lastly, let's try a word problem on differences of two squares. 

Example 7
Mr daniel's first son is 3 years older than his second son. If the son of their ages is 73. Calculate the differences of the squares of their ages

Let the first son age by $x$
Let the second son age be $y$

From the wordings of the first sentence
$x=3+y$

From the second sentence
$x+y=73$

Substituting the value of $x$ in $x+y=73$
$3+y+y=73$
$3+2y=73$
$2y=73-3$
$2y=70$

Divide both side by $2$
$\frac{2y}{2}=\frac{70}{2}$
$y=35$

Inserting the value of $y$ in $x=3+y$
$x=3+35$
$x=38$

The first son is $38$, the second son of Mr. Daniel is $35$.

But, we were asked to calculate the difference between the square of their ages
$(38)^2-(35)^2=(38+35)(38-35)$
                         $=(73)(3)$
                         $=219$

The difference between two squares is a very important mathematese. You will need the idea of the difference of two squares to factorize some quadratic expressions and surd expressions.
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