Theory:

Identities are the algebraic equation which is always true regardless of the values assigned to the variables.
The identities give an alternative method of solving problems on the multiplication of algebraic expressions and numbers.
Let us recall some of the squaring identities...
  • Identity I: a+b2=a2+2ab+b2
  • Identity II: ab2=a22ab+b2
  • Identity III: (a+b)(ab)=(a2b2)
  • Identity IV: (x+a)(x+b)=x2+(a+b)x+ab
 
Click here! To learn the derivation of the square identities.
 
Click here! To learn the expansion of some of the squared terms using the recalled identities.
Expansion of trinomial of the form \((a + b+ c)^{2}\)
Consider the expansion of the trinomial \((a + b+ c)^{2}\).
 
Replace \(x = a + b\) and \(y = c\) in the known identity \((x + y)^{2} = x^{2} + y^{2} +2xy\).
 
Then:
 
\((a + b+ c)^{2} = (a + b)^{2} + c^{2} +2(a + b)c\)
 
\(= a^{2} + b^{2} + 2ab + c^{2} +2ac +2bc\)
 
Therefore, we have:
 
Identity V: \((a + b+ c)^{2} = a^{2} + b^{2} + c^{2} +2ab +2bc+ 2ac \)
Example:
1. Find the expansion of \((2x + 3y +5z)^{2}\) using suitable identity.
 
Compare \((2x + 3y +5z)^{2}\) with \((a + b+ c)^{2}\).
 
Here, \(a = 2x\), \(b = 3y\) and \(c = 5z\).
 
So substitute the values of \(a\), \(b\) and \(c\) in the identity \((a + b+ c)^{2} = a^{2} + b^{2} + c^{2} +2ab +2bc+ 2ac \).
 
\((2x + 3y +5z)^{2} = (2x)^{2} + (3y)^{2} + (5z)^{2} +2(2x)(3y) +2(3y)(5z)+ 2(2x)(5z)\)
 
\(= 4x^{2} + 9y^{2} + 25z^{2} + 12xy + 30yz + 20xz\)
 
 
2. Find the expansion of \((3x - 3y + 4z)^{2}\) using suitable identity.
 
Compare \((3x - 3y + 4z)^{2}\) with \((a + b+ c)^{2}\).
 
Here, \(a = 3x\), \(b = - 3y\) and \(c = 4z\).
 
\((3x - 3y + 4z)^{2} = (3x)^{2} + (-3y)^{2} + (4z)^{2} + 2(3x)(-3y) +2(-3y)(4z)+ 2(3x)(4z)\)
 
\(= 9x^{2} + 9y^{2} + 16z^{2} - 18xy - 24yz + 24xz\)
 
 
3. Find the expansion of \((x - 3y - 4z)^{2}\) using suitable identity.
 
Compare \((x - 3y - 4z)^{2}\) with \((a + b+ c)^{2}\).
 
Here, \(a = x\), \(b = - 3y\) and \(c = - 4z\).
 
\((x - 3y - 4z)^{2} = (x)^{2} + (-3y)^{2} + (-4z)^{2} + 2(x)(-3y) +2(-3y)(-4z)+ 2(x)(-4z)\)
 
\(= x^{2} + 9y^{2} + 16z^{2} - 6xy + 24yz - 8xz\)