Question

Quais os valores de x para que a função f(x) x^2-5×+6 Seja positivo?

Answer 1

Olá Neuanderson.


Como queremos valores positivos, significa obter valores maiores que 0.

Organizando e achando os valores de x para que a função seja positiva:


$\mathsf{x^2-5x+6\ \textgreater \ 0}\\\\\\\mathsf{\Delta=(-5)^2-4\cdot1\cdot6}\\\mathsf{\Delta=25-24}\\\mathsf{\Delta=1}\\\\\\\mathsf{x=\dfrac{-(-5)\pm\sqrt{1}}{2\cdot1}}\\\\\\\mathsf{x^+=\dfrac{5+1}{2}\qquad\qquad\qquad\qquad x^-=\dfrac{5-1}{2}}\\\\\\\mathsf{x^+=\dfrac{6}{2}\qquad\qquad\qquad\qquad\quad ~x^-=\dfrac{4}{2}}\\\\\\\boxed{\mathsf{x^+=3}}\qquad\qquad\qquad\qquad\boxed{{\mathsf{x^-=2}}}$

$\mathsf{x^2-5x+6=(x-3)\cdot(x-2)}$

Temos uma inequação produto, precisamos então fazer a regra de sinais.


$\mathsf{x-3\qquad\qquad\qquad_{\underline{------}}~\underset\circ2~{_\underline{----}}~\underset\circ3~{_\underline{+++++}}_\big\blacktriangleright}}}\\\\\mathsf{x-2\qquad\qquad\qquad_{\underline{------}}~\underset\circ2~{_\underline{++++}}~\underset\circ3~{_\underline{+++++}}_\big\blacktriangleright}}}\\\\\mathsf{\cap~\quad\qquad\qquad\qquad_{\underline{++++++~}}~\underset\circ2~{_\underline{----}}~\underset\circ3~{_\underline{+++++}}_\big\blacktriangleright}}$

$\boxed{\mathsf{S=\{x\in\mathbb{R}~|~x\ \textless \ 2~ou~x\ \textgreater \ 3\}}}$


Dúvidas? comente.