Question

a) x² - 3x + 1 = 0
b) (x +6) (3 - x) = 20
c) -4x² + 4x + 3 = 0
d) x² - 4 - (x - 1)² = x² - 8
e) (x + 3)² = 2x (x + 7)
f) (x + 3/2) * (x + 1) = 2x² -11 24)
Resolva, em R, as quações.
a) x/x + 1 - x/ 1 - x = 8/3
b) 1/3 - x + 6/x² -9 = 2/x
c) 2/x² -1 - x/x -1 = 2
Ps: Preciso urgente!!!

Answer 1

Explicação:

23.

a) $x^2-3x+1=0$

$\Delta=(-3)^2-4\cdot1\cdot1$

$\Delta=9-4$

$\Delta=5$

$x=\dfrac{-(-3)\pm\sqrt{5}}{2\cdot1}=\dfrac{3\pm\sqrt{5}}{2}$

$x'=\dfrac{3+\sqrt{5}}{2}$

$x"=\dfrac{3-\sqrt{5}}{2}$

$S=\left\{\dfrac{3-\sqrt{5}}{2},\dfrac{3+\sqrt{5}}{2}\right\}$

b) $(x+6)\cdot(3-x)=20$

$-x^2-3x+18=20$

$x^2+3x+2=0$

$\Delta=3^2-4\cdot1\cdot2$

$\Delta=9-8$

$\Delta=1$

$x=\dfrac{-3\pm\sqrt{1}}{2\cdot1}=\dfrac{-3\pm1}{2}$

$x'=\dfrac{-3+1}{2}~\longrightarrow~x'=\dfrac{-2}{2}~\longrightarrow~x'=-1$

$x"=\dfrac{-3-1}{2}~\longrightarrow~x"=\dfrac{-4}{2}~\longrightarrow~x"=-2$

$S=\{-2,-1\}$

c) $-4x^2+4x+3=0$

$\Delta=4^2-4\cdot(-4)\cdot3$

$\Delta=16+48$

$\Delta=64$

$x=\dfrac{-4\pm\sqrt{64}}{2\cdot(-4)}=\dfrac{-4\pm8}{-8}$

$x'=\dfrac{-4+8}{-8}~\longrightarrow~x'=\dfrac{4}{-8}~\longrightarrow~x'=\dfrac{-1}{2}$

$x"=\dfrac{-4-8}{-8}~\longrightarrow~x"=\dfrac{-12}{-8}~\longrightarrow~x"=\dfrac{3}{2}$

$S=\left\{\dfrac{-1}{2},\dfrac{3}{2}\right\}$

d) $x^2-4-(x-1)^2=x^2-8$

$-4-x^2+2x-1=-8$

$x^2-2x-3=0$

$\Delta=(-2)^2-4\cdot1\cdot(-3)$

$\Delta=4+12$

$\Delta=16$

$x=\dfrac{-(-2)\pm\sqrt{16}}{2\cdot1}=\dfrac{2\pm4}{2}$

$x'=\dfrac{2+4}{2}~\longrightarrow~x'=\dfrac{6}{2}~\longrightarrow~x'=3$

$x"=\dfrac{2-4}{2}~\longrightarrow~x"=\dfrac{-2}{2}~\longrightarrow~x"=-1$

$S=\{-1,3\}$

e) $(x+3)^2=2x\cdot(x+7)$

$x^2+6x+9=2x^2+14x$

$x^2+8x-9=0$

$\Delta=8^2-4\cdot1\cdot(-9)$

$\Delta=64+36$

$\Delta=100$

$x=\dfrac{-8\pm\sqrt{100}}{2\cdot1}=\dfrac{-8\pm10}{2}$

$x'=\dfrac{-8+10}{2}~\longrightarrow~x'=\dfrac{2}{2}~\longrightarrow~x'=1$

$x"=\dfrac{-8-10}{2}~\longrightarrow~x"=\dfrac{-18}{2}~\longrightarrow~x"=-9$

$S=\{-9,1\}$

f) $\left(x+\dfrac{3}{2}\right)(x+1)=2x^2-11$

$x^2+x+\dfrac{3x}{2}+\dfrac{3}{2}=2x^2-11$

$2x^2+2x+3x+3=4x^2-22$

$2x^2-5x-25=0$

$\Delta=(-5)^2-4\cdot2\cdot(-25)$

$\Delta=25+200$

$\Delta=225$

$x=\dfrac{-(-5)\pm\sqrt{225}}{2\cdot2}=\dfrac{5\pm15}{4}$

$x'=\dfrac{5+15}{4}~\longrightarrow~x'=\dfrac{20}{4}~\longrightarrow~x'=5$

$x"=\dfrac{5-15}{4}~\longrightarrow~x"=\dfrac{-10}{4}~\longrightarrow~x"=\dfrac{-5}{2}$

$S=\left\{\dfrac{-5}{2},5\right\}$

24.

a) $\dfrac{x}{x+1}-\dfrac{x}{1-x}=\dfrac{8}{3}$

$\dfrac{x-x^2-x^2-x}{1-x^2}=\dfrac{8}{3}$

$-6x^2=8-8x^2$

$2x^2=8$

$x^2=4$

$x'=2$

$x"=-2$

$S=\{-2,2\}$

b) $\dfrac{1}{3-x}+\dfrac{6}{x^2-9}=\dfrac{2}{x}$

$\dfrac{-x-3+6}{x^2-9}=\dfrac{2}{x}$

$2x^2-18=-x^2+3x$

$3x^2-3x-18=0$

$x^2-x-6=0$

$\Delta=(-1)^2-4\cdot1\cdot(-6)$

$\Delta=1+24$

$\Delta=25$

$x=\dfrac{-(-1)\pm\sqrt{25}}{2\cdot1}=\dfrac{1\pm5}{2}$

$x'=\dfrac{1+5}{2}~\longrightarrow~x'=\dfrac{6}{2}~\longrightarrow~x'=3$ (não serve)

$x"=\dfrac{1-5}{2}~\longrightarrow~x"=\dfrac{-4}{2}~\longrightarrow~x"=-2$

$S=\{-2\}$

c) $\dfrac{2}{x^2-1}-\dfrac{x}{x-1}=2$

$\dfrac{2-x^2-x}{x^2-1}=2$

$2x^2-2=-x^2-x+2$

$3x^2+x-4=0$

$\Delta=1^2-4\cdot3\cdot(-4)$

$\Delta=1+48$

$\Delta=49$

$x=\dfrac{-1\pm\sqrt{49}}{2\cdot3}=\dfrac{-1\pm7}{6}$

$x'=\dfrac{-1+7}{6}~\longrightarrow~x'=\dfrac{6}{6}~\longrightarrow~x'=1$ (não serve)

$x"=\dfrac{-1-7}{6}~\longrightarrow~x"=\dfrac{-8}{6}~\longrightarrow~x"=\dfrac{-4}{3}$

$S=\left\{\dfrac{-4}{3}\right\}$