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a-2-b-2-15-2a-b-




Question Number 122405 by Khalmohmmad last updated on 16/Nov/20
a^2 −b^2 =15  2a−b=?
$${a}^{\mathrm{2}} −{b}^{\mathrm{2}} =\mathrm{15} \\ $$$$\mathrm{2}{a}−{b}=? \\ $$
Commented by Dwaipayan Shikari last updated on 16/Nov/20
Infintely many answers  a=(√(15)) ,b=0  2a−b=2(√(15))  a=4  b=1  2a−b=7  ....
$${Infintely}\:{many}\:{answers} \\ $$$${a}=\sqrt{\mathrm{15}}\:,{b}=\mathrm{0} \\ $$$$\mathrm{2}{a}−{b}=\mathrm{2}\sqrt{\mathrm{15}} \\ $$$${a}=\mathrm{4}\:\:{b}=\mathrm{1} \\ $$$$\mathrm{2}{a}−{b}=\mathrm{7} \\ $$$$…. \\ $$
Answered by ajfour last updated on 16/Nov/20
a=±(√(b^2 +15))  2a−b = −b±2(√(b^2 +15))  As one example for  integral solution  let  b=1  2a−b=−1±8 = 7, −9
$${a}=\pm\sqrt{{b}^{\mathrm{2}} +\mathrm{15}} \\ $$$$\mathrm{2}{a}−{b}\:=\:−{b}\pm\mathrm{2}\sqrt{{b}^{\mathrm{2}} +\mathrm{15}} \\ $$$${As}\:{one}\:{example}\:{for}\:\:{integral}\:{solution} \\ $$$${let}\:\:{b}=\mathrm{1} \\ $$$$\mathrm{2}{a}−{b}=−\mathrm{1}\pm\mathrm{8}\:=\:\mathrm{7},\:−\mathrm{9} \\ $$
Answered by mr W last updated on 16/Nov/20
(a−b)(a+b)=15=1×15=3×5   { ((a−b=1)),((a+b=15)) :}⇒a=8,b=7⇒2a−b=9   { ((a−b=−1)),((a+b=−15)) :}⇒...   { ((a−b=15)),((a+b=1)) :}⇒...   { ((a−b=−15)),((a+b=−1)) :}⇒...   { ((a−b=3)),((a+b=5)) :}⇒a=4,b=1⇒2a−b=7  ......
$$\left({a}−{b}\right)\left({a}+{b}\right)=\mathrm{15}=\mathrm{1}×\mathrm{15}=\mathrm{3}×\mathrm{5} \\ $$$$\begin{cases}{{a}−{b}=\mathrm{1}}\\{{a}+{b}=\mathrm{15}}\end{cases}\Rightarrow{a}=\mathrm{8},{b}=\mathrm{7}\Rightarrow\mathrm{2}{a}−{b}=\mathrm{9} \\ $$$$\begin{cases}{{a}−{b}=−\mathrm{1}}\\{{a}+{b}=−\mathrm{15}}\end{cases}\Rightarrow… \\ $$$$\begin{cases}{{a}−{b}=\mathrm{15}}\\{{a}+{b}=\mathrm{1}}\end{cases}\Rightarrow… \\ $$$$\begin{cases}{{a}−{b}=−\mathrm{15}}\\{{a}+{b}=−\mathrm{1}}\end{cases}\Rightarrow… \\ $$$$\begin{cases}{{a}−{b}=\mathrm{3}}\\{{a}+{b}=\mathrm{5}}\end{cases}\Rightarrow{a}=\mathrm{4},{b}=\mathrm{1}\Rightarrow\mathrm{2}{a}−{b}=\mathrm{7} \\ $$$$…… \\ $$

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