a plus b plus c WHOLE SQUARE FORMULA

In this section, we are going to see the formula or expansion  for (a + b + c)2.

That is, 

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

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

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

Example 1 : 

Expand : 

(5x + 3y + 2z)2

Solution : 

(5x + 3y + 2z)is in the form of (a + b + c)2

Comparing (a + b + c)2 and (5x + 3y + 2z)2, we get

a  =  5x

b  =  3y

c  =  2z

Write the formula / expansion for (a + b + c)2.

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

Substitute 5x for a, 3y for b and 2z for c. 

(5x + 3y + 2z)2  :

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

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

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

25x+ 9y+ 4z+ 30xy + 12yz + 20xz

Example 2 : 

If a + b + c  =  15 , ab + bc + ac  =  25, then find the value of

a2 + b2 + c2

Solution : 

To get the value of (a2 + b2 + c2), we can use the formula or expansion of (a + b + c)2.

Write the formula / expansion for (a + b + c)2.

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

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

Substitute 15 for (a + b + c)  and 25 for (ab + bc + ac).

(15)2  =  a2 + b+ c+ 2(25)

225  =  a2 + b+ c+ 50

Subtract 50 from each side. 

175  =  a2 + b+ c2

So, the value of a2 + b+ cis 175. 

a plus b minus c Whole Square Formula

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

The formula or expansion for (a + b + c)is

(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

Example :

Expand : 

(x + 2y - z)2

Solution : 

(x + 2y - z)is in the form of (a + b - c)2

Comparing (a + b - c)2 and (x + 2y - z)2, we get

a  =  x

b  =  2y

c  =  z

Write the formula / expansion for (a + b - c)2.

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

Substitute x for a, 2y for b and z for c. 

(x + 2y - z):

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

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

So, the expansion of (x + 2y - z)is  

x+ 4y+ z+ 4xy - 4yz - 2xz

a minus b plus c Whole Square Formula

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

The formula or expansion for (a + b + c)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

Example :

Expand : 

(3x - y + 2z)2

Solution : 

(3x - y + 2z)is in the form of (a - b + c)2

Comparing (a + b - c)2 and (3x - y + 2z)2, we get

a  =  3x

b  =  y

c  =  2z

Write the formula / expansion for (a - b + c)2.

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

Substitute 3x for a, y for b and 2z for c. 

(3x - y + 2z):

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

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

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

 9x+ y+ 4z- 6xy - 4yz + 12xz

a minus b minus c Whole Square Formula

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

The formula or expansion for (a + b + c)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 band 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

Example :

Expand : 

(x - 2y - 3z)2

Solution : 

(x - 2y - 3z)is in the form of (a - b - c)2

Comparing (a - b - c)2 and (x - 2y - 3z)2, we get

a  =  x

b  =  2y

c  =  3z

Write the formula / expansion for (a - b - c)2.

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

Substitute x for a, 2y for b and 3z for c. 

(x - 2y - 3z):

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

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

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

x+ 4y+ 9z- 4xy + 12yz - 6xz

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