Question

If $P=7{{x}^{2}}+5xy-9{{y}^{2}}$ , $Q=4{{y}^{2}}-3{{x}^{2}}$ and $R=4{{x}^{2}}+xy+5{{y}^{2}}$ show that $P+Q+R=8{{x}^{2}}+6xy$ .

Answer 1

Hint: For answering this question we will use the given values $P=7{{x}^{2}}+5xy-9{{y}^{2}}$ , $Q=4{{y}^{2}}-3{{x}^{2}}$ and $R=4{{x}^{2}}+xy+5{{y}^{2}}$. And derive the required expression which we need to prove $P+Q+R=8{{x}^{2}}+6xy$ by rearranging the like terms aside and performing all the arithmetic operations. We will start by considering the LHS of the equation and then reach the RHS.

Complete step-by-step answer:
Now considering from the question we have $P=7{{x}^{2}}+5xy-9{{y}^{2}}$ , $Q=4{{y}^{2}}-3{{x}^{2}}$ and $R=4{{x}^{2}}+xy+5{{y}^{2}}$ .
We have to prove $P+Q+R=8{{x}^{2}}+6xy$ .
After adding the given three expressions we will have $P+Q+R=7{{x}^{2}}+5xy-9{{y}^{2}}+4{{y}^{2}}-3{{x}^{2}}+4{{x}^{2}}+xy+5{{y}^{2}}$ .
By rearranging all the ${{x}^{2}}$ terms aside we will have $P+Q+R=\left( 7+4-3 \right){{x}^{2}}+5xy-9{{y}^{2}}+4{{y}^{2}}+xy+5{{y}^{2}}$ .
By rearranging all the $xy$ terms aside we will have
$P+Q+R=\left( 7+4-3 \right){{x}^{2}}+\left( 5+1 \right)xy-9{{y}^{2}}+5{{y}^{2}}+4{{y}^{2}}$ .
By rearranging all the ${{y}^{2}}$ terms aside we will have
$P+Q+R=\left( 7+4-3 \right){{x}^{2}}+\left( 5+1 \right)xy+\left( -9+4+5 \right){{y}^{2}}$ .
After performing the addition operation we have for the term ${{x}^{2}}$ we will have $P+Q+R=\left( 11-3 \right){{x}^{2}}+\left( 5+1 \right)xy+\left( -9+4+5 \right){{y}^{2}}$ .
After performing all the operations related to the term ${{x}^{2}}$ we will have $P+Q+R=8{{x}^{2}}+\left( 5+1 \right)xy+\left( -9+4+5 \right){{y}^{2}}$ .
After performing all the operations we have for the term $xy$ we will have $P+Q+R=8{{x}^{2}}+6xy+\left( -9+4+5 \right){{y}^{2}}$ .
After performing all the operations related to the term ${{y}^{2}}$ we will have $P+Q+R=8{{x}^{2}}+6xy+\left( 0 \right){{y}^{2}}$ .
Now we have the conclusion that the expression $P+Q+R=8{{x}^{2}}+6xy$ where $P=7{{x}^{2}}+5xy-9{{y}^{2}}$ , $Q=4{{y}^{2}}-3{{x}^{2}}$ and $R=4{{x}^{2}}+xy+5{{y}^{2}}$ .
Hence it is proved that the expression is valid.

Note: While answering questions of this type we should be careful while substituting all the given values and rearranging the terms and performing the arithmetic operations. For example in case if we had a mistake and taken the wrong values like $Q=4{{y}^{2}}-3xy$ instead of $Q=4{{y}^{2}}-3{{x}^{2}}$ we will have $P+Q+R=11{{x}^{2}}+3xy$ which is not the required expression which we need to prove.