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Question Number 213751 by hardmath last updated on 15/Nov/24

$$\mathrm{m};\mathrm{n}\in {\mathbb{Z}}_{+}$$$${2\mathrm{m}}^2+{\mathrm{n}}^2-\mathrm{mn}=54$$$$$$$$1.(\mathrm{m};\mathrm{n})=?$$$$2.(\mathrm{m};\mathrm{n})=?$$$$................$$

Commented by hardmath last updated on 15/Nov/24

$$\mathrm{a})1\mathrm{b})2\mathrm{c})3\mathrm{d})4\mathrm{e})5$$

Commented by hardmath last updated on 18/Nov/24

$${2\mathrm{m}}^2-{\mathrm{n}}^2-\mathrm{mn}=54$$$$\mathrm{solution}\left(\mathrm{please}\right)$$

Commented by mehdee7396 last updated on 19/Nov/24

$$(m-n)(m+n)+m(m-n)=54$$$$(m-n)(2m+n)=54$$$$m-n=1\mathrm{\&}2m+n=54\times$$$$m-n=2\mathrm{\&}2m+n=27\times$$$$m-n=3\mathrm{\&}2m+n=18\Rightarrow m=7\mathrm{\&}n=4✓$$$$m-n=6\mathrm{\&}2m+n=9\Rightarrow m=5\mathrm{\&}m=1✓$$$$$$

Answered by Ghisom last updated on 15/Nov/24

$$\mathrm{no}\mathrm{solution}\mathrm{for}m,n\in {\mathbb{Z}}^{+}$$

Commented by hardmath last updated on 15/Nov/24

$$\mathrm{m};\mathrm{n}\in \mathbb{N}$$

Commented by Ghisom last updated on 16/Nov/24

$$\mathrm{no}\mathrm{solution}\mathrm{for}m,n\in \mathbb{N}$$

Answered by mehdee7396 last updated on 16/Nov/24

$$2m^2-nm+n^2-54=0$$$$△=9(48-n^2)isdosenothaveacompletroot$$$$\Rightarrow \nexists m,n\in \mathbb{N}∍2m^2+n^2-mn=54$$$$$$

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