Question :-

Factorise the following equations ;

• 1. 9x² + 12xy + 4y²

• 2. 16x² + 8xy + y²

• 3. 16x² - 8xy + y²

Answer :-

Given :-

Equations ;

• 1. 9x² + 12xy + 4y²

• 2. 16x² + 8xy + y²

• 3. 16x² - 8xy + y²

Required to find :-

• Factorise the given equations ?

Identities used :-

• ( a + b )² = a² + 2ab + b²

• ( a - b )² = a² - 2ab + b²

Solution :-

1.

9x² + 12xy + 4y²

( 3x)² + 2 ( 3x )( 2y ) + ( 2y )²

This is in the form of ;

• ( a + b )² = a² + 2ab + b²

Here,

a = 3x

b = 2y

This implies ;

=> ( 3x + 2y)²

2.

16x² + 8xy + y²

( 4x )² + 2 ( 4x )( y ) + ( y )²

This is in the form of ;

• ( a + b )² = a² + 2ab + b²

Here,

a = 4x

b = y

This implies ;

=> ( 4x + y )²

3.

16x² - 8xy + y²

( 4x )² - 2 ( 4x )( y ) + ( y )²

This is in the form of ;

• ( a - b )² = a² - 2ab + b²

Here,

a = 4x

b = y

This implies ;

=> ( 4x - y )²

Therefore ;

The given quadratic equations in 2 variables had been factorised as follows ;

• 1. 9x² + 12xy + 4y² = ( 3x + 2y )²

• 2. 16x² + 8xy + y² = ( 4x + y )²

• 3. 16x² - 8xy + y² = ( 4x - y )²

Additional Information :-

How to verify whether the factorisation is correct or wrong ?

Answer :-

To verify whether the factorisation is correct or wrong for that we should multiply the factors which we got by factorisation .

So,

Let me explain you with an example .

Example :-

From the above it is clear that ;

9x² + 12xy + 4y² = ( 3x + 2y )²

So

Multiplying ( 3x + 2y ) twice will gives us the equation BACK .

Let's prove it ;

( 3x + 2y ) ( 3x + 2y ) = 9x² + 12xy + 4y²

3x ( 3x + 2y ) + 2y ( 3x + 2y ) = 9x² + 12xy + 4y²

9x² + 6xy + 6xy + 4y² = 9x² + 12xy + 4y²

9x² + 12xy + 4y² = 9x² + 12xy + 4y²

LHS = RHS

Hence verified ✔️

This trick is very useful when DEALING with COMPLICATED factorisations .