Question

${\text{a}}^2{\text{+b}}^2\text{is equal to ?}$

Answer 1

To solve this we use some identities in some steps :

Step 1: Here we use $({\text{a+b)}}^2{\text{=a}}^2{\text{+b}}^2\text{+2ab}$

${\text{a}}^2{\text{+b}}^2{\text{=(a+b)}}^2\text{-2ab}$(after shifting the sides)

Step 2: Now we use the identity $({\text{a-b)}}^2{\text{=a}}^2{\text{+b}}^2\text{-2ab}$

${\text{a}}^2{\text{+b}}^2{\text{=(a-b)}}^2\text{+2ab}$

Step 3: Now we add the equation obtained in these two steps

$$\begin{gathered} {\text{a}}^2{\text{+b}}^2{\text{+a}}^2{\text{+b}}^2{\text{=(a+b)}}^2{\text{-2ab+(a-b)}}^2\text{+2ab} \\ 2{\text{a}}^2{\text{+2b}}^2{\text{=(a+b)}}^2{\text{+(a-b)}}^2 \\ {\text{2(a}}^2{\text{+b}}^2{\text{)=(a+b)}}^2{\text{+(a-b)}}^2 \\ {\text{(a}}^2{\text{+b}}^2\text{)=}\frac{{\text{(a+b)}}^2{\text{+(a-b)}}^2}{2} \end{gathered}$$

Hence, we get $({\text{a}}^2{\text{+b}}^2\text{)=}\frac{{\text{(a+b)}}^2{\text{+(a-b)}}^2}{2}$.