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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.
