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Question Number 213751 by hardmath last updated on 15/Nov/24
m ; n ∈ Z_+ 2m^2 + n^2 − mn = 54 1. (m;n)=? 2. (m;n)=? ................
$$\mathrm{m}\:;\:\mathrm{n}\:\in\:\mathbb{Z}_{+} \\ $$$$\mathrm{2m}^{\mathrm{2}} \:+\:\mathrm{n}^{\mathrm{2}} \:−\:\mathrm{mn}\:=\:\mathrm{54} \\ $$$$ \\ $$$$\mathrm{1}.\:\left(\mathrm{m};\mathrm{n}\right)=? \\ $$$$\mathrm{2}.\:\left(\mathrm{m};\mathrm{n}\right)=? \\ $$$$……………. \\ $$
Commented by hardmath last updated on 15/Nov/24
a) 1 b) 2 c) 3 d) 4 e) 5
$$\left.\mathrm{a}\left.\right)\left.\:\left.\mathrm{1}\left.\:\:\:\:\:\mathrm{b}\right)\:\mathrm{2}\:\:\:\:\:\mathrm{c}\right)\:\mathrm{3}\:\:\:\:\:\mathrm{d}\right)\:\mathrm{4}\:\:\:\:\:\mathrm{e}\right)\:\mathrm{5} \\ $$
Commented by hardmath last updated on 18/Nov/24
2m^2 − n^2 − mn = 54 solution (please)
$$\mathrm{2m}^{\mathrm{2}} \:−\:\mathrm{n}^{\mathrm{2}} \:−\:\mathrm{mn}\:=\:\mathrm{54} \\ $$$$\mathrm{solution}\:\left(\mathrm{please}\right) \\ $$
Commented by mehdee7396 last updated on 19/Nov/24
(m−n)(m+n)+m(m−n)=54 (m−n)(2m+n)=54 m−n=1 & 2m+n=54 × m−n=2 & 2m+n=27 × m−n=3 & 2m+n=18 ⇒m=7 &n=4 ✓ m−n=6 & 2 m+n=9⇒m=5 & m=1 ✓
$$\left({m}−{n}\right)\left({m}+{n}\right)+{m}\left({m}−{n}\right)=\mathrm{54} \\ $$$$\left({m}−{n}\right)\left(\mathrm{2}{m}+{n}\right)=\mathrm{54} \\ $$$${m}−{n}=\mathrm{1}\:\:\&\:\:\mathrm{2}{m}+{n}=\mathrm{54}\:\:× \\ $$$${m}−{n}=\mathrm{2}\:\:\&\:\:\mathrm{2}{m}+{n}=\mathrm{27}\:× \\ $$$${m}−{n}=\mathrm{3}\:\:\&\:\:\mathrm{2}{m}+{n}=\mathrm{18}\:\:\Rightarrow{m}=\mathrm{7}\:\&{n}=\mathrm{4}\:\:\checkmark \\ $$$${m}−{n}=\mathrm{6}\:\:\&\:\mathrm{2}\:{m}+{n}=\mathrm{9}\Rightarrow{m}=\mathrm{5}\:\&\:{m}=\mathrm{1}\:\checkmark\:\: \\ $$$$ \\ $$
Answered by Ghisom last updated on 15/Nov/24
no solution for m, n ∈Z^+
$$\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{m},\:{n}\:\in\mathbb{Z}^{+} \\ $$
Commented by hardmath last updated on 15/Nov/24
m;n∈N
$$\mathrm{m};\mathrm{n}\in\mathbb{N} \\ $$
Commented by Ghisom last updated on 16/Nov/24
no solution for m, n ∈N
$$\mathrm{no}\:\mathrm{solution}\:\mathrm{for}\:{m},\:{n}\:\in\mathbb{N} \\ $$
Answered by mehdee7396 last updated on 16/Nov/24
2m^2 −nm+n^2 −54=0 △=9(48−n^2 ) is dosenot have a complet root ⇒∄m,n∈N ∍ 2m^2 +n^2 −mn=54
$$\mathrm{2}{m}^{\mathrm{2}} −{nm}+{n}^{\mathrm{2}} −\mathrm{54}=\mathrm{0} \\ $$$$\bigtriangleup=\mathrm{9}\left(\mathrm{48}−{n}^{\mathrm{2}} \right)\:\:\:{is}\:{dosenot}\:{have}\:{a}\:{complet}\:{root}\:\: \\ $$$$\Rightarrow\nexists{m},{n}\in\mathbb{N}\:\backepsilon\:\mathrm{2}{m}^{\mathrm{2}} +{n}^{\mathrm{2}} −{mn}=\mathrm{54} \\ $$$$ \\ $$

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