Question

Determine o valor de w para que f(x) = x² + (w - 6)x + 4 tenha duas raízes iguais.

Answer 1

A função admite duas raízes reais iguais se e somente se ∆=0.

$\mathsf{f(x)=x^2+(w-6)x+4}\\\mathsf{a=1~~~b=w-6~~~c=4}\\\mathsf{\Delta=b^2-4ac}\\\mathsf{(w-6)^2-4\cdot\,1\cdot\,4}\\\mathsf{\Delta=(w-6)^2-16}\\\mathsf{\Delta=0\implies(w-6)^2-16=0}\\\mathsf{(w-6)^2=16}\\\mathsf{(w-6)=\pm\sqrt{16}}\\\mathsf{(w-6)=\pm4\implies\,w=6\pm4}\\\begin{cases}\mathsf{w_{1}=6+4=10}\\\mathsf{w_{2}=6-4=2}\end{cases}$