# DIFFERENCE OF TWO SQUARES

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.

Related post
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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