Question Number 218138 by hardmath last updated on 30/Mar/25
$$\sqrt[{\mathrm{4}}]{\mathrm{629}\:−\:\mathrm{x}}\:\:+\:\:\sqrt[{\mathrm{4}}]{\mathrm{77}\:\:+\:\:\mathrm{x}}\:\:=\:\:\mathrm{8} \\ $$$$\mathrm{Find}:\:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$
Answered by Ghisom last updated on 30/Mar/25
$$\mathrm{easy}\:\mathrm{to}\:\mathrm{see}: \\ $$$$\mathrm{629}\:\mathrm{is}\:\mathrm{almost}\:\mathrm{5}^{\mathrm{4}} ,\:\mathrm{77}\:\mathrm{is}\:\mathrm{almost}\:\mathrm{3}^{\mathrm{4}} \\ $$$$\mathrm{629}−\mathrm{4}=\mathrm{5}^{\mathrm{4}} \\ $$$$\mathrm{77}+\mathrm{4}=\mathrm{3}^{\mathrm{4}} \\ $$$$\Rightarrow\:{x}_{\mathrm{1}} =\mathrm{4} \\ $$$$\mathrm{harder}\:\mathrm{to}\:\mathrm{see}: \\ $$$$\mathrm{629}−\mathrm{548}=\mathrm{3}^{\mathrm{4}} \\ $$$$\mathrm{77}+\mathrm{548}=\mathrm{5}^{\mathrm{4}} \\ $$$$\Rightarrow\:{x}_{\mathrm{2}} =\mathrm{548} \\ $$$$\mathrm{no}\:\mathrm{other}\:\mathrm{solution} \\ $$
Answered by mehdee7396 last updated on 31/Mar/25
![((629−x))^(1/4) =u & ((77+x))^(1/4) =v u^4 =629−x & 77+x=v^4 u^4 +v^4 =706 ; u+v=s=8 & uv=p [(u+v)^2 −2uv]^2 −2(uv)^2 =706 p^2 −128p+1695=0 p=64±49⇒p=15 or 113 if p=15 & s=8 u^2 −8u+15=0 u=4±1⇒u=5 & v=3 ⇒x=4 ✓ or u=3⇒v= 5⇒x=548✓ if p=113 & s=8 u^2 −8u+113=0 u=4±(√(97))i⇒u=4+(√(97))i & v=−4+(√(97))i ⇒x=... or x=...](https://www.tinkutara.com/question/Q218141.png)
$$\sqrt[{\mathrm{4}}]{\mathrm{629}−{x}}={u}\:\:\:\&\:\:\sqrt[{\mathrm{4}}]{\mathrm{77}+{x}}={v} \\ $$$${u}^{\mathrm{4}} =\mathrm{629}−{x}\:\:\:\&\:\:\:\mathrm{77}+{x}={v}^{\mathrm{4}} \\ $$$${u}^{\mathrm{4}} +{v}^{\mathrm{4}} =\mathrm{706}\:\:\:;\:\:{u}+{v}={s}=\mathrm{8}\:\:\:\&\:\:{uv}={p} \\ $$$$\left[\left({u}+{v}\right)^{\mathrm{2}} −\mathrm{2}{uv}\right]^{\mathrm{2}} −\mathrm{2}\left({uv}\right)^{\mathrm{2}} =\mathrm{706} \\ $$$${p}^{\mathrm{2}} −\mathrm{128}{p}+\mathrm{1695}=\mathrm{0} \\ $$$${p}=\mathrm{64}\pm\mathrm{49}\Rightarrow{p}=\mathrm{15}\:{or}\:\:\mathrm{113} \\ $$$${if}\:\:{p}=\mathrm{15}\:\:\&\:{s}=\mathrm{8} \\ $$$${u}^{\mathrm{2}} −\mathrm{8}{u}+\mathrm{15}=\mathrm{0} \\ $$$${u}=\mathrm{4}\pm\mathrm{1}\Rightarrow{u}=\mathrm{5}\:\&\:{v}=\mathrm{3}\:\:\Rightarrow{x}=\mathrm{4}\:\checkmark \\ $$$${or}\:\:{u}=\mathrm{3}\Rightarrow{v}=\:\mathrm{5}\Rightarrow{x}=\mathrm{548}\checkmark \\ $$$${if}\:\:{p}=\mathrm{113}\:\:\:\&\:\:\:{s}=\mathrm{8} \\ $$$${u}^{\mathrm{2}} −\mathrm{8}{u}+\mathrm{113}=\mathrm{0} \\ $$$${u}=\mathrm{4}\pm\sqrt{\mathrm{97}}{i}\Rightarrow{u}=\mathrm{4}+\sqrt{\mathrm{97}}{i}\:\&\:{v}=−\mathrm{4}+\sqrt{\mathrm{97}}{i} \\ $$$$\Rightarrow{x}=…\:\:\:\:{or}\:\:\:\:{x}=… \\ $$$$ \\ $$
Commented by Hanuda354 last updated on 31/Mar/25
$${x}\:=\:\mathrm{4}\:\mathrm{or}\:\:{x}\:=\:\mathrm{548} \\ $$