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If-3-2-a-b-c-a-b-c-then-a-bc-




Question Number 136340 by liberty last updated on 21/Mar/21
If (√(3+(√2))) = (√((a+(√b))/c)) + (√((a−(√b))/c))  then a+bc = ?
$${If}\:\sqrt{\mathrm{3}+\sqrt{\mathrm{2}}}\:=\:\sqrt{\frac{{a}+\sqrt{{b}}}{{c}}}\:+\:\sqrt{\frac{{a}−\sqrt{{b}}}{{c}}} \\ $$$${then}\:{a}+{bc}\:=\:? \\ $$
Answered by Ñï= last updated on 21/Mar/21
(√(3+(√2)))=(√x)+(√y)  3+(√2)=x+y+2(√(xy))  ⇒ { ((3=x+y)),(((√2)=2(√(xy)))) :}  ⇒x=((3+(√7))/2)      y=((3−(√7))/2)  ⇒a=3     b=7      c=2  a+bc=17
$$\sqrt{\mathrm{3}+\sqrt{\mathrm{2}}}=\sqrt{{x}}+\sqrt{{y}} \\ $$$$\mathrm{3}+\sqrt{\mathrm{2}}={x}+{y}+\mathrm{2}\sqrt{{xy}} \\ $$$$\Rightarrow\begin{cases}{\mathrm{3}={x}+{y}}\\{\sqrt{\mathrm{2}}=\mathrm{2}\sqrt{{xy}}}\end{cases} \\ $$$$\Rightarrow{x}=\frac{\mathrm{3}+\sqrt{\mathrm{7}}}{\mathrm{2}}\:\:\:\:\:\:{y}=\frac{\mathrm{3}−\sqrt{\mathrm{7}}}{\mathrm{2}} \\ $$$$\Rightarrow{a}=\mathrm{3}\:\:\:\:\:{b}=\mathrm{7}\:\:\:\:\:\:{c}=\mathrm{2} \\ $$$${a}+{bc}=\mathrm{17} \\ $$

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