Question

Determine as raízes da equação :
a) 3x2
– 8x = 0
b) -2y2
+ 32 = 0
c) r
2
+2r + 3 = 0
d) 100 x
2
+60x – 27 = 0
e) 5 x
2
- 45 = 0
f) x
2
+8x + 16 = 49
g) (x – 6 )2
= 2( x + 18 )
h) t(t+2) = 7t
i) x
2
+12X +32 = 0
j) x
2
-4X -32 = 0
k) x
2
– 2X -3 = 0
l) x
2
– 16 =0
m) x
2
+ 4X – 32 = 0
n) 1,5 x2
+ 0,1x = 0,6
o) x² - 5x + 6 = 0
p) x² - 8x + 12 = 0
q) x² + 2x - 8 = 0
r) x² - 5x + 8 = 0
s) 2x² - 8x + 8 = 0
t) x² - 4x - 5 = 0
u) -x² + x + 12 = 0
v) x
2
- 3x - 4 = 0

Answer 1

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$\tt a)~\sf3x^2-8x=0\\\sf x(3x-8)=0\\\sf x=0\\\sf 3x-8=0\\\sf 3x=8\\\sf x=\dfrac{8}{3}$

$\tt b)~\sf -2y^2+32=0\\\sf2y^2=32\\\sf y^2=\dfrac{32}{2}\\\sf y^2=16\\\sf y=\pm\sqrt{16}\\\sf y=\pm4$

$\tt c)~\sf r^2+2r+3=0\\\sf \Delta=4-12=-8<0\implies \not\exists~x~\in\mathbb{R}$

$\tt d)~\sf 100x^2+60x-27=0\\\sf \Delta=3600+10800=14400\\\sf x=\dfrac{-60\pm120}{200}\begin{cases}\sf x_1=\dfrac{-60+120}{200}=\dfrac{60\div20}{200\div20}=\dfrac{3}{10}\\\sf x_2=\dfrac{-60-120}{200}=-\dfrac{180\div20}{200\div20}=-\dfrac{9}{10}\end{cases}$

$\tt e)~\sf 5x^2-45=0\\\sf 5x^2=45\\\sf x^2=\dfrac{45}{5}\\\sf x^2=9\\\sf x=\pm\sqrt{9}\\\sf x=\pm-3$

$\tt f)~\sf x^2+8x+16=49\\\sf (x+4)^2=49\\\sf x+4=\pm\sqrt{49}\\\sf x+4=\pm7\\\sf x+4=7\\\sf x=7-4\\\sf x=3\\\sf x+4=-7\\\sf x=-4-7\\\sf x=-11$

$\tt g)~\sf (x-6)^2=2(x+18)\\\sf x^2-12x+\diagdown\!\!\!\!\!\!36=2x+\diagdown\!\!\!\!\!\!36\\\sf x^2-12x-2x=0\\\sf x^2-14x=0\\\sf x(x-14)=0\\\sf x=0\\\sf x-14=0\\\sf x=14$

$\tt h)~\sf t(t+2)=7t\\\sf t^2+2t-7t=0\\\sf t^2-5t=0\\\sf t(t-5)\\\sf t=0\\\sf t-5=0\\\sf t=5$

$\tt i)~\sf x^2+12+32=0\\\sf\Delta=144-128=16\\\sf x=\dfrac{-12\pm4}{2}\begin{cases}\sf x_1=\dfrac{-12+4}{2}=-\dfrac{8}{2}=-4\\\sf x_2=\dfrac{-12-4}{2}=-\dfrac{16}{2}=-8\end{cases}$

$\tt j)~\sf x^2-4x-32=0\\\sf \Delta=16+128=144\\\sf x=\dfrac{4\pm12}{2}\begin{cases}\sf x_1=\dfrac{4+12}{2}=\dfrac{16}{2}=8\\\sf x_2=\dfrac{4-12}{2}=-\dfrac{8}{2}=-4\end{cases}$

$\tt l)~\sf x^2-16=0\\\sf x^2=16\\\sf x=\pm\sqrt{16}\\\sf x=\pm4$

$\tt m)~\sf x^2+4x-32=0\\\sf\Delta=16+128=144\\\sf x=\dfrac{-4\pm12}{2}\begin{cases}\sf x_1=\dfrac{-4+12}{2}=\dfrac{8}{2}=4\\\sf x_2=\dfrac{-4-12}{2}=-\dfrac{16}{2}=-8\end{cases}$

$\tt n)~\sf1,5x^2+0,1x=0,6\cdot10\\\sf 15x^2+x=6\\\sf15x^2+x-6=0\\\sf\Delta=1+360=361\\\sf x=\dfrac{-1\pm19}{60}\begin{cases}\sf x_1=\dfrac{-1+19}{60}=\dfrac{18\div6}{60\div6}=\dfrac{3}{10}\\\sf x_2=\dfrac{-1-19}{60}=-\dfrac{20\div20}{60\div20}=-\dfrac{1}{3}\end{cases}$

$\tt o)~\sf x^2-5x+6=0\\\sf\Delta=25-24=1\\\sf x=\dfrac{5\pm1}{2}\begin{cases}\sf x_1=\dfrac{5+1}{2}=\dfrac{6}{2}=3\\\sf x_2=\dfrac{5-1}{2}=\dfrac{4}{2}=2\end{cases}$

$\tt p)~\sf x^2-8x+12=0\\\sf\Delta=64-48=16\\\sf x=\dfrac{8\pm4}{2}\begin{cases}\sf x_1=\dfrac{8+4}{2}=\dfrac{12}{2}=6\\\sf x_2=\dfrac{8-4}{2}=\dfrac{4}{2}=2\end{cases}$

$\tt q)~\sf x^2+2x-8=0\\\sf\Delta=4+32=36\\\sf x=\dfrac{-2\pm6}{2}\begin{cases}\sf x_1=\dfrac{-2+6}{2}=\dfrac{4}{2}=2\\\sf x_2=\dfrac{-2-6}{2}=-\dfrac{8}{2}=-4\end{cases}$

$\tt r)~\sf x^2-5x+8=0\\\sf\Delta=25-32=-7\implies \not\exists~x~\in~\mathbb{R}$

$\tt s)~\sf 2x^2-8x+8=0\div2\\\sf x^2-4x+4=0\\\sf (x-2)^2=0\\\sf x-2=0\\\sf x=2$

$\tt t)~\sf x^2-4x-5=0\\\sf\Delta=16+20=36\\\sf x=\dfrac{4\pm6}{2}\begin{cases}\sf x_1=\dfrac{4+6}{2}=\dfrac{10}{2}=5\\\sf x_2=\dfrac{4-6}{2}=-\dfrac{2}{2}=-1\end{cases}$

$\tt u)~\sf -x^2+x+12=0\cdot(-1)\\\sf x^2-x-12=0\\\sf\Delta=1+48=49\\\sf x=\dfrac{1\pm7}{2}\begin{cases}\sf x_1=\dfrac{1+7}{2}=\dfrac{8}{2}=4\\\sf x_2=\dfrac{1-7}{2}=-\dfrac{6}{2}=-3\end{cases}$

$\tt v)~\sf x^2-3x-4=0\\\sf\Delta=9+16=25\\\sf x=\dfrac{3\pm5}{2}\begin{cases}\sf x_1=\dfrac{3+5}{2}=\dfrac{8}{2}=4\\\sf x_2=\dfrac{3-5}{2}=-\dfrac{2}{2}=-1\end{cases}$