Question

factorise the following using appropriate identities
i)9x²+6xy+y²
ii)4y²-4y+1
iii)x²-(y/100)²
​

Answer 1

$\huge{\blue {\underline{\underline{\tt{\orange{ANSWER}}}}}}$

Given :

$\longrightarrow{\tt{Given \; Quadratic\; Expression\;:-}}$

$\mathsf{1. \;\;\; 9{x}^{2}+6xy+{y}^{2}}{\leftarrow}$

$\mathsf{2. \;\;\; 4 {y}^{2} - 4y + 1}{\leftarrow}$

$\mathsf{3. \;\;\; {x}^{2} - (\dfrac{y}{100} {)}^{2} }{\leftarrow}$

$\rule{400}{1}$

Required to find :

1. Factorised form of the given quadratic expressions .

$\rule{400}{1}$

Explanation :

What is a Factorisation ?

Factorisation is the method of expanding or reducing the size of an polynomial expression into its factors .

When we multiply this factors again we get back the polynomial expression .

Factorisation doesn't depend upon the type of an expression .

We can factorise any type of expression such as

• Quadratic expression
• Cubic expression
• etc.

For factorisation of an expression we can use many methods .

Some of the methods are ;

1. By using identity .

2. By splitting the middle term .

3. By performing long division and factorisation of the quotient .

These are some of methods used in factorisation .

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Solution :

❶

$\longrightarrow{\tt{Factorise\; 9{x}^{2}+6xy+{y}^{2}}}$

Consider the given quadratic expression

$\tt{9{x}^{2}+6xy+{y}^{2}}$

$\tt{{(3x)}^{2} + 2(3x)(2)+{(y)}^{2}}$

Here, compare this expression with an Identity .

The identity is ;

$\mathsf{{(x+y)}^{2} = {x}^{2} + 2(x)(y) + {y}^{2} }$

Hence, the factorised form of the given expression is ;

$\tt{\red{\large{\implies{{(3x + y)}^{2}}}}}$

$\rule{400}{4}$

❷

$\longrightarrow{\tt{Factorise \; 4{y}^{2} - 4y + 1}}$

Consider the given quadratic expression

$\tt{4{y}^{2} - 4y + 1}$

$\tt{(2{y)}^{2} - 2(2y)(1) + (1{)}^{2}}$

Here, compare this expression with an identity .

The identity is ;

$\mathsf{{(x-y)}^{2} = {x}^{2} - 2(x)(y) + {y}^{2} }$

Hence, the factorised form of the given expression is ;

$\tt{\blue{\large{\implies{(2y - 1{)}^{2}}}}}$

$\rule{400}{4}$

❸

$\longrightarrow{\tt{Factorise \; {x}^{2} - \dfrac{{y}^{2}}{100}}}$

Consider the given expression

$\tt{ {x}^{2} - \dfrac{{y}^{2}}{100}}$

$\tt{{(x)}^{2} - (\dfrac{y}{10}{)}^{2}}$

Here, compare this expression with an identity .

The identity is ;

$\mathsf{(x + y)(x - y) = {x}^{2} - {y}^{2}}$

Hence, the factorised form of the given expression is ;

$\tt{\green{\large{\implies{(x + \dfrac{y}{10})(x - \dfrac{y}{10})}}}}$

$\rule{400}{4}$

✅ Hence solved .