Question

1) Calcule as raízes, caso existam, das funções a seguir:
a) f(x) = x2 + 5x + 4
b) f(x) = -x2 + 3x – 2

c) f(x) = -x2 + 6x – 9
d) f(x) = x2 – 4x + 5

Answer 1

$\Large\boxed{\begin{array}{l}\tt a)~\sf f(x)=x^2+5x+4\\\sf x^2+5x+4=0\\\sf s=-\dfrac{5}{1}=-5\\\sf p=\dfrac{4}{1}=4\\\sf x_1=-4~e~x_2=-1\\\sf pois~(-4)+(-1)=-5\\\sf e~(-4)\cdot(-1)=4\\\sf (-4,0)~(-1,0)\end{array}}$

$\Large\boxed{\begin{array}{l}\tt b)~\sf f(x)=-x^2+3x-2\\\sf -x^2+3x-2=0\\\sf s=-\dfrac{3}{-1}=3\\\\\sf p=\dfrac{-2}{-1}=2\\\sf x_1=1~e~x_2=2~pois\\\sf 1+2=3~e~1\cdot2=2\\\sf (1,0)~(2,0)\end{array}}$

$\Large\boxed{\begin{array}{l}\tt c)~\sf f(x)=-x^2+6x-9\\\sf -x^2+6x-9=0\\\sf s=-\dfrac{6}{-1}=6\\\sf p=\dfrac{-9}{-1}=9\\\sf x_1=x_2=3\\\sf pois~3+3=6~e~3\cdot3=9\\\sf (3,0)\end{array}}$

$\Large\boxed{\begin{array}{l}\tt d)~\sf f(x)=x^2-4x+5\\\sf x^2-4x+5=0\\\sf\Delta=b^2-4ac\\\sf\Delta=(-4)^2-4\cdot1\cdot5\\\sf\Delta=16-20\\\sf\Delta=-4<0\implies \not\exists x\in\mathbb{R}\\\sf portanto~n\tilde ao~possui~ra\acute izes\end{array}}$