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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