Question

qual é o valor da equação 2 cos^2 x + 3 cos x +1 =0

Answer 1

Resposta:

$2cos^{2}x+3cosx+1=0\\ Chamando de y=cosx, temos: \\2y^{2}+3y+1=0\\\\Aplicando~a~f\'{o}rmula~de~Bhaskara~para~2y^{2}+3y+1=0~~\\e~comparando~com~(a)x^{2}+(b)x+(c)=0,~temos~a=2{;}~b=3~e~c=1\\\\\Delta=(b)^{2}-4(a)(c)=(3)^{2}-4(2)(1)=9-(8)=1\\\\y^{'}=\frac{-(b)-\sqrt{\Delta}}{2(a)}=\frac{-(3)-\sqrt{1}}{2(2)}=\frac{-3-1}{4}=\frac{-4}{4}=-1\\\\y^{''}=\frac{-(b)+\sqrt{\Delta}}{2(a)}=\frac{-(3)+\sqrt{1}}{2(2)}=\frac{-3+1}{4}=\frac{-2}{4}=-0,5$

$Para~y^{'}=-1\\y^{'}=y=-1 \Rightarrow y=cosx=-1 \Rightarrow x=arccos(-1) \Rightarrow x=\pi\\\\S=\{x\in \mathbb{R}; x=\pm\pi+2k\pi, k\in\mathbb{Z}\}\\\\Para~y^{''}=-0,5\\y^{''}=y=-0,5 \Rightarrow y=cosx=-0,5 \Rightarrow x=arccos(-0,5) \Rightarrow x=\frac{2\pi}{3} \\\\S=\{x\in \mathbb{R}; x=\pm\frac{2\pi}{3}+2k\pi, k\in\mathbb{Z}\}$