Question

Simplify \[{a^2} + {b^2} = ?\]

Answer 1

Hint: We use two formulas which on expansion give similar terms that are given in the question. Adding or subtracting the required values we find the value required in the question.
* Square of sum of two numbers ‘a’ and ‘b’ is given by \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
* Square of difference of two numbers ‘a’ and ‘b’ is given by \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]

Complete step by step answer:
We have to find the value of \[{a^2} + {b^2}\]
We know square of sum of two numbers ‘a’ and ‘b’ is given by \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
We add or subtract the required values in this formula to obtain the value of \[{a^2} + {b^2}\]
Since, \[{(a + b)^2} = {a^2} + {b^2} + 2ab\]
We subtract \[2ab\] from both sides of the equation.
\[ \Rightarrow {(a + b)^2} - 2ab = {a^2} + {b^2} + 2ab - 2ab\]
Cancel the terms having the same magnitude but opposite sign i.e. \[2ab\] and \[ - 2ab\] from the RHS of the equation.
\[ \Rightarrow {(a + b)^2} - 2ab = {a^2} + {b^2}\]
\[ \Rightarrow {a^2} + {b^2} = {(a + b)^2} - 2ab\]

\[\therefore \] The value of \[{a^2} + {b^2}\] is \[{(a + b)^2} - 2ab\].

Note:
Students might try to simplify the term given in the question by substituting different values of ‘a’ and ‘b’ in the equation. But since we are not given anything about the nature of the elements ‘a’ and ‘b’, we cannot limit the numbers which can be substituted in their place. Also, it will be a long process as we can substitute any value. But we have to find a general simplification of the formula.
Alternate method:
We know square of difference of two numbers ‘a’ and ‘b’ is given by \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]
We add or subtract the required values in this formula to obtain the value of \[{a^2} + {b^2}\]
Since, \[{(a - b)^2} = {a^2} + {b^2} - 2ab\]
We add \[2ab\] to both sides of the equation.
\[ \Rightarrow {(a - b)^2} + 2ab = {a^2} + {b^2} - 2ab + 2ab\]
Cancel the terms having the same magnitude but opposite sign i.e. \[2ab\] and \[ - 2ab\] from the RHS of the equation.
\[ \Rightarrow {(a - b)^2} + 2ab = {a^2} + {b^2}\]
\[ \Rightarrow {a^2} + {b^2} = {(a - b)^2} + 2ab\]

\[\therefore \] The value of \[{a^2} + {b^2}\] is \[{(a - b)^2} + 2ab\]