a2 + b2 is equal to?

Answer:

We need to find (a2 + b2)

We know by identity that (a2 + b2)= [(a + b)2 + (a – b)2]/2

Let us consider the equation

(a + b)2 = a2 + b2 + 2ab……………………….(1)

Thus,

a2 + b2 = (a + b)2 – 2ab………………………..(2)

Also consider

(a – b)2 = a2 + b2 – 2ab………………………..(3)

Thus,

a2 + b2 = (a – b)2 + 2ab……………………….(4)

Addition of equation (1) and (2) we get

(a + b)2 + (a – b)2 = [a2 + b2 + 2ab] + [a2 + b2 – 2ab]…………….(5)

(a + b)2 + (a – b)2 = 2a2 + 2b2

(a + b)2 + (a – b)2 = 2(a2 + b2)

(a2 + b2) = [(a + b)2 + (a – b)2]/2…………………..(6)

Let us prove the above equation.

Consider a = 2 and b= 3, substitute in the equation (6) we get

(22 + 32) = [(2 + 3)2 +(2 – 3)2]/2

Consider LHS,

LHS = (22 + 32)

LHS = (4 + 9)

LHS = 13

Consider RHS

RHS = [(2 + 3)2 +(2 – 3)2]/2

RHS = [(52) + (-1)2]/2

RHS = [25 + 1]/2

RHS = 13

Formula

(a2 + b2) = [(a + b)2 + (a – b)2]/2

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