Question

calcule o valor de m na equação:
x²-6+m=0
a) as raízes sejam mais e diferentes A>0
b) as raízes sejam reais e iguais A=0
c) as raízes não sejam reais A<0

Answer 1

$\mathsf{x^2-6+m=0~~~~(equa\c{c}\~ao~do~segundo~grau~incompleta.)}$

$\mathsf{Coeficientes:}\begin{Bmatrix}\mathsf{a=1}\\\mathsf{b=0}\\\mathsf{c=-6+m}\end.\\\\\\\mathsf{\Delta=b^2-4ac}\\\\\mathsf{\Delta=0^2-4\cdot1\cdot(-6+m)}\\\\\mathsf{\Delta=-4\cdot(-6+m)}\\\\\mathsf{\Delta=24-4m}}\\\\\\\mathsf{Para~duas~ra\'izes~reais~e~distintas~o~discriminante~(\Delta)~deve~ser~maior~que~zero.}}\\\\\\\mathsf{24-4m\ \textgreater \ 0}\\\\\mathsf{-4m\ \textgreater \ -24}\\\\\mathsf{4m\ \textless \ 24}\\\\\mathsf{m\ \textless \ \dfrac{24}{4}}\\\\\mathsf{m\ \textless \ 6}$

$\mathsf{Para~duas~ra\'izes~reais~e~iguais~o~discriminante~(\Delta)~deve~ser~igual~a~zero.}}\\\\\\\mathsf{24-4m=0}\\\\\mathsf{-4m=-24}\\\\\mathsf{4m=24}\\\\\mathsf{m=\dfrac{24}{4}}\\\\\mathsf{m=6}\\\\\\\mathsf{Para~que~a~equa\c{c}\~ao~n\~ao~possua~ra\'izes~reais~o~discriminante~(\Delta)~deve}}\\\mathsf{ser~menor~que~zero.}\\\\\\\mathsf{24-4m\ \textless \ 0}\\\\\mathsf{-4m\ \textless \ -24}\\\\\mathsf{4m\ \textgreater \ 24}\\\\\mathsf{m\ \textgreater \ \dfrac{24}{4}}\\\\\mathsf{m\ \textgreater \ 6}$