EXPANDING BRACKETS TO THE POWER OF 2

Expansion of Binomials with Power 2

(a + b)²  =  (a + b)(a + b)

(a + b)²  =  a² + ab + ab + b²

(a + b)²  =  a² + 2ab+ b²

(a - b)²  =  (a - b)(a - b)

(a - b)²  =  a² - ab - ab + b²

(a - b)²  =  a² - 2ab+ b²

Expansion of Trinomials with Power 2

Expansion of (a + b + c)² :

(a + b + c)²  =  (a + b + c)(a + b + c)

(a - b)²  =  a² + ab + ac + ab + b² + bc + ac + bc + c²

(a - b)²  =  a² + b² + c² + 2ab + 2bc + 2ac

Expansion of (a + b - c)² :

To get expansion for (a + b - c)², let us consider the expansion of (a + b + c)². 

(a + b + c)²  =  a² + b² + c² + 2ab + 2bc + 2ac

In (a + b + c)², if c is negative, then we have 

(a + b - c)²

In the terms of the expansion for (a + b + c)², consider the terms in which we find 'c'.

They are c², bc, ca.

Even if we take negative sign for 'c' in c², the sign of c² will be positive.  Because it has even power 2. 

The terms bc, ac will be negative. Because both 'b' and 'a' are multiplied by 'c' that is negative.  

Finally, we have 

(a + b - c)²  =  a² + b² + c² + 2ab - 2bc - 2ac

Expansion of (a - b + c)² :

To get expansion for (a - b + c)², let us consider the expansion of (a + b + c)². 

The expansion of (a + b + c)² is

(a + b + c)²  =  a² + b² + c² + 2ab + 2bc + 2ca

In (a + b + c)², if b is negative, then we have 

(a - b + c)²

In the terms of the expansion for (a + b + c)², consider the terms in which we find "b".

They are b², ab, bc.

Even if we take negative sign for 'b' in b², the sign of b² will be positive.  Because it has even power 2. 

The terms ab, bc will be negative. Because both 'a' and 'c' are multiplied by 'b' that is negative.  

Finally, we have 

(a - b + c)²  =  a² + b² + c² - 2ab - 2bc + 2ac

Expansion of (a - b - c)² :

To get the expansion of (a - b - c)², let us consider the expansion of (a + b + c)². 

The expansion of (a + b + c)² is

(a + b + c)²  =  a² + b² + c² + 2ab + 2bc + 2ca

In (a + b + c)², if b and c are negative, then we have 

(a - b - c)²

In the terms of the expansion for (a + b + c)², consider the terms in which we find 'b' and 'c'.

They are b², c², ab, bc, ac.

Even if we take negative sign for 'b' in b² and negative sign for 'c' in c², the sign of both b² and c² will be positive.  Because they have even power 2. 

The terms 'ab' and 'ac' will be negative.

Because, in 'ab', 'a' is multiplied by "b" that is negative. 

Because, in 'ac', 'a' is multiplied by "c" that is negative.  

The term 'bc' will be positive.

Because, in 'bc', both 'b' and 'c' are negative.    

That is,

negative ⋅ negative  =  positive  

Finally, we have 

(a - b - c)²  =  a² + b² + c² - 2ab + 2bc - 2ac

Expansion of Trinomials with Power 2 - Summary

(a + b + c)²  =  a² + b² + c² + 2ab + 2bc + 2ca

(a + b - c)²  =  a² + b² + c² + 2ab - 2bc - 2ca

(a - b + c)²  =  a² + b² + c² - 2ab - 2bc + 2ca

(a - b - c)²  =  a² + b² + c² - 2ab + 2bc - 2ca

Instead of memorizing all the above expansions, we may memorize the first expansion and apply values of b and c along with signs.

Expanding Brackets to the Power of 2 - Practice Questions

Question 1 :

Expand :

(5x + 3)²

Solution :

(5x + 3)²  =  (5x)² + 2(5x)(3) + 3²

(5x + 3)²  =  25x² + 30x + 9

Question 2 :

Expand :

(3x - 2)²

Solution :

(3x - 2)²  =  (3x)² - 2(3x)(2) + 2²

(3x - 2)²  =  9x² - 12x + 4

Question 3 :

Find the value of 105².

Solution :

105²  =  (100 + 5)²

105²  =  100² + 2(100)(5) + 5²

105²  =  10000 + 1000 + 25

105²  =  11025

Question 4 :

Find the value of 99².

Solution :

99²  =  (100 - 1)²

99²  =  100² - 2(100)(1) + 1²

99²  =  10000 - 200 + 1

99²  =  9801

Question 5 : 

Expand : 

(5x + 3y + 2z)²

Solution : 

(5x + 3y + 2z)²  :

= (5x)² + (3y)² + (2z)² + 2(5x)(3y) + 2(3y)(2z) + 2(5x)(2z)

(5x + 3y + 2z)²  =  25x² + 9y² + 4z² + 30xy + 12yz + 20xz

So, the expansion of (5x + 3y + 2z)² is  

25x² + 9y² + 4z² + 30xy + 12yz + 20xz

Question 6 :

Expand : 

(x + 2y - z)²

Solution : 

(x + 2y - z)² :

=  x² + (2y)² + z² + 2(x)(2y) - 2(2y)(z) - 2(x)(z)

(x + 2y - z)²  =  x² + 4y² + z² + 4xy - 4yz - 2xz

So, the expansion of (x + 2y - z)² is  

x² + 4y² + z² + 4xy - 4yz - 2xz

Question 7 :

Expand : 

(3x - y + 2z)²

Solution : 

(3x - y + 2z)² :

=  (3x)² + y² + (2z)² - 2(3x)(y) - 2(y)(2z) + 2(3x)(2z)

(3x - y + 2z)²  =  9x² + y² + 4z² - 6xy - 4yz + 12xz

So, the expansion of (3x - y + 2z)² is 

 9x² + y² + 4z² - 6xy - 4yz + 12xz

Question 8 :

Expand : 

(x - 2y - 3z)²

Solution : 

(x - 2y - 3z)² :

=  x² + (2y)² + (3z)² - 2(x)(2y) + 2(2y)(3z) - 2(x)(3z)

(x - 2y - 3z)²  =  x² + 4y² + 9z² - 4xy + 12yz - 6xz

So, the expansion of (x - 2y - 3z)² is 

x² + 4y² + 9z² - 4xy + 12yz - 6xz

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