question_answer 11)

(a) What should be added to \[{{x}^{2}}+xy+{{y}^{2}}\] to obtain \[2{{x}^{2}}+3xy\]? (b)  What should be subtracted from \[2a+8b+10\] to get \[-3a+7b+16\]?

Answer:

                (a) Required expression                 \[=2{{x}^{2}}+3xy-({{x}^{2}}+xy+{{y}^{2}})\]                 \[=2{{x}^{2}}+3xy-{{x}^{2}}-xy-{{y}^{2}}\]                 \[=2{{x}^{2}}-{{x}^{2}}+3xy-xy-{{y}^{2}}\]                                            \[\left| \text{rearranging terms} \right.\]                                \[={{x}^{2}}+2xy-{{y}^{2}}\].                 (b) Required expression                 \[=(2a+8b+10)-(-3a+7b+16)\]                 \[=2a+8b+10+3a-7b-16\]                 \[=2a+3a+8b-7b+10-16\]                                              \[\left| \text{rearranging terms} \right.\] \[=5a+b-6\] What should be taken away from \[3{{x}^{2}}-4{{y}^{2}}+5xy+20\] to obtain \[-{{x}^{2}}-{{y}^{2}}+6xy\]\[+20\]?                  Required expression \[=(3{{x}^{2}}-4{{y}^{2}}+5xy+20)-(-{{x}^{2}}-{{y}^{2}}+6xy+20)\] \[=3{{x}^{2}}-4{{y}^{2}}+5xy+20+{{x}^{2}}+{{y}^{2}}-6xy-20\] \[=3{{x}^{2}}+{{x}^{2}}+{{y}^{2}}-4{{y}^{2}}+5xy-6xy+20-20\]     \[\left| \text{rearranging terms} \right.\] \[=4{{x}^{2}}-3{{y}^{2}}-xy\].