Question

Arătați ca, oricare ar fi m€R, următoarele ecuații au soluții reale:
a)mx^2-2x-m+2=0
b)x^2-2x-m^2-2m=0

Answer 1

$a)~m x^{2} -2x-m+2=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ m( x^{2} -1)-2(x-1)=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ m(x-1)(x+1)-2(x-1)=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ (x-1)[m(x+1)-2]=0. \\ \boxed{Ecuatia~admite~solutia~x=1~indiferent~de~valoarea~lui~m.} \\ *O~alta~solutie~ar~fi~x= \frac{2}{m} -1,~dar~aceasta~nu~este~valabila~pentru \\ m=0.*$

$b)~ x^{2} -2x- m^{2} -2m=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ ( x^{2} - m^{2} )-2(x+m)=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ (x-m)(x+m)-2(x+m)=0\ \textless \ =\ \textgreater \ \\ \ \textless \ =\ \textgreater \ (x+m)(x-m-2)=0. \\ \boxed{Ecuatia~admite~solutiile~x=-m~sau~x=m+2.}$