Question

determinati numerele reale m si n pentru care ecuatiile (x-2)^2+8=3x si (m-n)x^2-(m+n)x+2m+n+1=0 sunt echivalente​

Answer 1

 

$\displaystyle\bf\\E_1)\\(x-2)^2+8=3x\\x^2-4x+4-3x+8=0\\x^2-7x+12=0\\\\E_2)\\(m-n)x^2-(m+n)x+2m+n+1=0\\\\E_1~si~E_2~sunt~echivalente~daca~se~indeplinesc~conditiile\!:\\\\1)~Coeficientul~lui~x^2~din~E_1~=~coeficientul~lui~x^2~din~E_2\\2)~Coeficientul~lui~x~din~E_1~=~coeficientul~lui~x~din~E_2\\3)~Termenul~liber~din~E_1=termenul~liber~din~E_2$

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$\displaystyle\bf\\m-n=1\\m+n=7\\2m+n+1=12\\\\Avem~3~ecuatii~cu~2~necunoscute.\\Rezolvam~primele~2~ecuatii~si~verificam~pe~a~3-a\\\\m-n=1\\m+n=7\\------+\\2m~~/~=8\\\\m=\frac{8}{2}\\\\ \boxed{\bf m=4}\\\\m+n=7\\n=7-m=7-4\\\boxed{\bf n=3}\\\\Verificam~ecuatia~a~3-a:\\2m+n+1=12\\2\cdot4+3+1=8+3+1=11+1=12~~Corect$