Question

56. Resolvam as seguintes inequações do 2º grau em R: -2x² - x + 1 < 0​

Answer 1

$\boxed{\begin{array}{l}\sf -2x^2-x+1<0\\\sf fac_{\!\!,}a~f(x)=-2x^2-x+1\\\sf queremos~os~valores~de~x~tais~que~f(x)<0.\\\sf ra\acute izes:\\\sf -2x^2-x+1=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-1)^2-4\cdot(-2)\cdot1\\\sf\Delta=1+8\\\sf\Delta=9\\\sf x=\dfrac{-b\pm\sqrt{\Delta}}{2a}\\\sf x=\dfrac{-(-1)\pm\sqrt{9}}{2\cdot(-2)}\\\sf x=\dfrac{1\pm3}{-4}\begin{cases}\sf x_1=\dfrac{1+3}{-4}=-\dfrac{4}{4}=-1\\\sf x_2=\dfrac{1-3}{-4}=\dfrac{2\div2}{4\div2}=\dfrac{1}{2}\end{cases}\end{array}}$

$\Large\boxed{\begin{array}{l}\underline{\rm estudo~do~sinal:}\\\sf f(x)>0\implies -1<x<\dfrac{1}{2}\\\sf f(x)<0\implies x<-1~~ou~~x>\dfrac{1}{2}\\\underline{\rm soluc_{\!\!,}\tilde ao~da~inequac_{\!\!,}\tilde ao:}\\\sf S=\bigg\{ x\in\mathbb{R}/x<-1~ou~x>\dfrac{1}{2}\bigg\}\end{array}}$