[tex]\text{Ecuatia de gradul 2, de forma: } \\ ax^2 +bx+c=0 \\ \text{poate fi scrisa si sub forma: } \\ x^2-Sx+P=0\;\;\;\;\;\;\;unde: \\ S=x_1+x_2\;\;\;\;\;(suma \;radacinilor) \\ P=x_1x_2\;\;\;\;\;(produsul \;radacinilor) \\ \\ a) \\ x^2+(m-5)x-3=0, \;\;\;cu\;conditia:\;\; x_1+x_2=4 \\ x_1+x_2=S = -(m-5) \\ =\ \textgreater \ \;\;5-m=4 \\ =\ \textgreater \ \;\;\;m=5-4=\boxed{1}[/tex]
[tex]b) \\ mx^2+(m+1)x-2=0\;\;\;cu\;conditia:\;\;\;x_1+x_2-x_1x_2 = - \frac{3}{4} \\ \text{Impartim ecuatia la }\;\; "m" \\ x^2+ \frac{m+1}{m}x- \frac{2}{m} =0 \\ \\ x_1+x_2=S=-\frac{m+1}{m} \\ \\ x_1x_2=P=- \frac{2}{m} \\ \\ =\ \textgreater \ \;\;x_1+x_2-x_1x_2 = S-P = -\frac{m+1}{m} -(- \frac{2}{m}) \\ \\ -\frac{m+1}{m} -(- \frac{2}{m})=- \frac{3}{4} \\ \\ -\frac{m+1+2}{m}=- \frac{3}{4}\;\;\;\;|*(-1)[/tex]
[tex]\frac{m+3}{m}= \frac{3}{4} \\ \\ 4(m+3)=3m \\ 4m+12=3m \\ 4m-3m=-12 \\ m=\boxed{-12}[/tex]
