Question

What is the formula of ${{x}^{2}}+{{y}^{2}}$?

Answer 1

Hint: For solving this question you should know about the whole square of two variables summation and subtraction. In this problem we will first write the formula of the whole square of summation of two variables and then we will find the rest part which is not provided in the question that will be the original answer for this and then we write again the whole part here as ${{\left( a\pm b \right)}^{2}}$ and subtract the part which is original answer for this.

Complete step by step answer:
So, according to the question we have to find the formula for finding the value of ${{x}^{2}}+{{y}^{2}}$. As we know that ${{\left( a+b \right)}^{2}}$ is written as ${{a}^{2}}+2ab+{{b}^{2}}$ and if we take the ${{\left( a+b \right)}^{2}}$ and the $2ab$ at one side and the rest part will be the answer providing for this part. So, if we use the same method for our question, then,
$\begin{align}
  & \Rightarrow {{\left( x+y \right)}^{2}}={{x}^{2}}+2xy+{{y}^{2}} \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{\left( x+y \right)}^{2}}-2xy \\
\end{align}$
This is the answer for ${{x}^{2}}+{{y}^{2}}$ in terms of ${{\left( x+y \right)}^{2}}$. And if we see the method for ${{\left( x-y \right)}^{2}}$, then as we know that ${{\left( a-b \right)}^{2}}$ is written as ${{a}^{2}}-2ab+{{b}^{2}}$ and if we take the ${{\left( a+b \right)}^{2}}$ to one side then the rest part is the solution. Thus if we repeat the same process for ${{\left( x-y \right)}^{2}}$, then we can write it as,
$\Rightarrow {{\left( x-y \right)}^{2}}={{x}^{2}}-2xy+{{y}^{2}}$
And if we add $2xy$ to both sides, then we get,
$\begin{align}
  & \Rightarrow {{\left( x-y \right)}^{2}}+2xy={{x}^{2}}+{{y}^{2}} \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{\left( x-y \right)}^{2}}+2xy\ldots \ldots \ldots \left( i \right) \\
 & \Rightarrow {{x}^{2}}+{{y}^{2}}={{\left( x+y \right)}^{2}}-2xy\ldots \ldots \ldots \left( ii \right) \\
\end{align}$
So, these both (i) and (ii) are the formulas of ${{x}^{2}}+{{y}^{2}}$.

Note: While solving these types of questions you have to keep in mind that the values of these can be generated directly by just making the sum change in the formulas. And these changes are very minor but we have to be careful about these because if any sign goes wrong, then the whole solution will be wrong.