Question

determine as raízes reais, quando houver, de cada equação polinomial do segundo grau incompleta. a)5x^2 - 10 = 0 b)10y^2 - 200y = 0 c)3x^2 + 27 = 0 d)30n^2 + 270n = 0 e)x^2 + x/5 = 0 f)2p^2 = 0 g)(5a - 1)^2 = 1 h)(2y + 3)^2 - 9 = 12y i)(5x + 3)/3 - 3 = (x^2 - 14)/7 j)(x - 1) . (x + 2) = x - 4 me ajudem

Answer 1

Resposta:

$a)~5x^{2} -10=0\\ \\ a=5,~~b=0,~~c=-10\\ \\ x=\sqrt{\frac{-c}{a} } \\ \\ x=\sqrt{\frac{-(-10)}{5} } =\sqrt{\frac{10}{5} } =+-\sqrt{2} \\ \\ S~~\left \{ {{-\sqrt{2},~~\sqrt{2} } {}} \}$

$b)~10y^{2} -200y=0\\ \\ a=10,~~b=-200,~~c=0\\ \\ x'=0\\ \\ x''=\frac{-b}{a} =\frac{-(-200)}{10} =\frac{200}{10} =20\\ \\ S~~\left \{ {{0,~~20} {}} \}$

$c)~3x^{2} +27=0\\ \\ a=3,~~b=0,~~c=27\\ \\ x=\sqrt{\frac{-c}{a} } =\sqrt{\frac{-27}{3} } =\sqrt{-9}$

Não possui nenhuma raiz real

$d)~30n^{2} +270n=0\\ \\ a=30,~~b=27,~~c=0\\ \\ x'=0\\ \\ x''=\frac{-b}{a} =\frac{-27}{30} =-\frac{9}{10} \\ \\ S~~\left \{ {{0,~-\frac{9}{10} } {}} \}$

$e)~x^{2} +\frac{x}{5} =0~~.(5)=\\ \\ 5x^{2} +x=0\\ \\ a=5,~~b=1,~~c=0\\ \\ x'=0\\ \\ x''=\frac{-b}{a} =\frac{-1}{5} =-\frac{1}{5} \\ \\ S~~\left \{ {{0,~-\frac{1}{5} } {}} \}$

$f)~2p^{2} =0\\ \\ a=2,~~b=0,~~c=0\\ \\ x=0$

$g)~(5a-1)^{2} =1\\ \\ (5a)^{2} -2.5a.1+1^{2} =1\\ \\ 25a^{2} -10a+1-1=0\\ \\ 25a^{2} -10a=0\\ \\ a=25,~~b=-10,~~c=0\\ \\ x'=0\\ \\ x''=\frac{-b}{a} =\frac{-(-10)}{25} =\frac{10}{25} =\frac{2}{5} \\ \\ S~~\left \{ {{0,~\frac{2}{5} } {}} \}$

$h)~(2y+3)^{2} -9=12y\\ \\ (2y)^{2} +2.2y.3+3^{2} -9=12y\\ \\ 4y^{2} +12y+9-9-12y=0\\ \\ 4y^{2} =0\\ \\ y=0$

$i)~\frac{5x+3}{3} -3=\frac{(x^{2}-14) }{7} \\ \\mmc=21\\ \\ \frac{7(5x+3)-63=3(x^{2}-14) }{21} \\ \\ 35x+21-63=3x^{2} -42\\ \\ 3x^{2} -35x-42-21+63=0\\ \\ 3x^{2} -35x=0\\ \\ a=3,~~b=-35,~~c=0\\ \\ x'=0\\ \\ x''=\frac{-b}{a} =\frac{-(-35)}{3} =\frac{35}{3} \\ \\ S~~\left \{ {{0,~\frac{35}{3} } {}} \}$

$j)~(x-1).(x+2)=x-4\\ \\ x^{2} +2x-x-2=x-4\\ \\ x^{2} +x-2=x-4\\ \\ x^{2} -x+x-2+4=0\\ \\ x^{2} +2=0\\ \\ a=1,~~b=0,~~c=2\\ \\ x=\sqrt{\frac{-c}{a} } \\ \\ x=\sqrt{\frac{-2}{1} } \\ \\ x=\sqrt{-1}$o correto é $\sqrt{-2}$

Não existe nenhuma raiz real