Simplify a2 + b2

Q. Simplify a2 + b2 = ?

The identity a2 + b2  is derived from the formula (a+b)2 = a2 + b2 + 2ab.

This simple formula is also used in proving The Pythagorean Theorem. Pythagoras’  Theorem is one of the first proofs in Mathematics.

Now, taking the equation (a+b)2 = a2 + b2 + 2ab

• To find the result of a2 + b2  we’ll be required to shift +2ab from RHS to LHS.

 So we have the equation, (a+b)2 – 2ab = a2 + b2 

 Therefore we get, a2 + b2  = (a+b)2 – 2ab 

Proving the equation a2 + b2  = (a+b)2 – 2ab 

• Taking the RHS

= (a + b) ² – 2ab

• Expanding the equation we get,

= (a+ b) (a + b) – 2ab

= a2  + ab + ab + b2 – 2ab

= a2  + 2ab + b2 – 2ab

= a2 + b2 + 2ab – 2ab by the Commutative Property of Addition.

= a2  + b2 + 2ab + (- 2ab) by the Definition of Subtraction.

a2  + b2  +[2ab + (- 2ab)] by the Associative Property of Addition.

= a² + b² + 2[0] by the Additive Inverse Property.

• The +2ab and the -2ab term cancel out, leaving us with: = a² + b² + 0 by the Multiplication Property of Zero.

= a² + b²  — hence proved.