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If-a-b-3-Find-a-2-b-2-2a-2b-a-2-b-2-4a-4-

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Question Number 190392 by Shrinava last updated on 02/Apr/23
If a + b = 3 Find: ((a^2 + b^2 − 2a − 2b)/(a^2 − b^2 − 4a + 4))
$$\mathrm{If}\:\:\:\mathrm{a}\:+\:\mathrm{b}\:=\:\mathrm{3} \\ $$$$\mathrm{Find}:\:\:\:\frac{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} \:−\:\mathrm{2a}\:−\:\mathrm{2b}}{\mathrm{a}^{\mathrm{2}} \:−\:\mathrm{b}^{\mathrm{2}} \:−\:\mathrm{4a}\:+\:\mathrm{4}} \\ $$
Commented by mokys last updated on 05/Apr/23
((a^2 +b^2 −2(a+b))/((a−b)(a+b)−4a+4)) = ((a^2 +(3−a)^2 −6)/(3(a−b)−4a+4)) = ((2a^2 +9−6a−6)/(3(2a−3)−4a+4))= ((2a^2 −6a+3)/(2a−5)) = ((2ab + 3)/(2(a−3)+1))= ((2ab+3)/(2b+1))
$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −\mathrm{2}\left({a}+{b}\right)}{\left({a}−{b}\right)\left({a}+{b}\right)−\mathrm{4}{a}+\mathrm{4}}\:=\:\frac{{a}^{\mathrm{2}} +\left(\mathrm{3}−{a}\right)^{\mathrm{2}} −\mathrm{6}}{\mathrm{3}\left({a}−{b}\right)−\mathrm{4}{a}+\mathrm{4}} \\ $$$$ \\ $$$$=\:\frac{\mathrm{2}{a}^{\mathrm{2}} +\mathrm{9}−\mathrm{6}{a}−\mathrm{6}}{\mathrm{3}\left(\mathrm{2}{a}−\mathrm{3}\right)−\mathrm{4}{a}+\mathrm{4}}=\:\frac{\mathrm{2}{a}^{\mathrm{2}} −\mathrm{6}{a}+\mathrm{3}}{\mathrm{2}{a}−\mathrm{5}} \\ $$$$ \\ $$$$=\:\frac{\mathrm{2}{ab}\:+\:\mathrm{3}}{\mathrm{2}\left({a}−\mathrm{3}\right)+\mathrm{1}}=\:\frac{\mathrm{2}{ab}+\mathrm{3}}{\mathrm{2}{b}+\mathrm{1}} \\ $$$$ \\ $$
Answered by Frix last updated on 02/Apr/23
? b=3−a ((a^2 +b^2 −2a−2b)/(a^2 −b^2 −4a+4))=((2a^2 −6a+3)/(2a−5))
$$? \\ $$$${b}=\mathrm{3}−{a} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −\mathrm{2}{a}−\mathrm{2}{b}}{{a}^{\mathrm{2}} −{b}^{\mathrm{2}} −\mathrm{4}{a}+\mathrm{4}}=\frac{\mathrm{2}{a}^{\mathrm{2}} −\mathrm{6}{a}+\mathrm{3}}{\mathrm{2}{a}−\mathrm{5}} \\ $$

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