Question

23) Resolva, em R as equações.
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

Answer 1

Explicação passo-a-passo:

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)

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

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

Answer 2

23.a) x² - 3x + 1 = 0

∆ = (-3)² - 4

∆ = 9 - 4

∆ = 5

x' = (3 - √5)/2

x" = (3 + √5)/2

b) (x + 6) (3 - x) = 20

3x - x² + 18 - 6x - 20 = 0

x² + 3x + 2 = 0

∆ = 3² - 4 × 2

∆ = 9 - 8

∆ = 1

x = (-3 ± √1)/2

x = (-3 ± 1)/2

x' = -4/2 = -2

x" = -2/2 = -1

c) -4x² + 4x + 3 = 0

∆ = 4² - 4 × -4 × 3

∆ = 16 + 48

∆ = 64

x = (-4 ± √64)/(2 × -4)

x = (-4 ± 8)/-8

x = (-1 ± 2)/-2

x' = (-1 - 2)/-2 = 3/2

x" = (-1 + 2)/-2 = -1/2

d) x² - 4 - (x - 1)² = x² - 8

x² - 2x - 3 = 0

∆ = (-2)² - 4 × (-3)

∆ = 4 + 12

∆ = 16

x = (2 ± √16)/2

x = 1 ± 2

x' = 1 - 2 = -1

x" = 1 + 2 = 3

e) (x + 3)² = 2x (x + 7)

x² + 8x - 9 = 0

∆ = 8² - 4 × (-9)

∆ = 64 + 36

∆ = 100

x = (-8 ± √100)/2

x = (-8 ± 10)/2

x = -4 ± 5

x' = -4 - 5 = -9

x" = -4 + 5 = 1

f) (x + 3/2)(x + 1) = 2x² -11

2x² - 5x - 25 = 0

∆ = (-5)² - 4 × 2 × (-25)

∆ = 25 + 200

∆ = 225

x = (5 ± √ 225)/(2 × 2)

x = (5 ± 15)/4

x' = -10/4 = -5/2

x" = 20/4 = 5

24. a) x/(x + 1) - x/(1 - x) = 8/3

(x - x² - x² - x)/(1 - x²) = 8/3

3(-2x²) = 8(1 - x²)

-6x² = 8 - 8x²

8x² - 6x² = 8

2x² = 8

x² = 4

x = ±√4

x' = -2

x" = 2

b) 1/(3 - x) + 6/(x² - 9) = 2/x

2x² - 18 = -x² + 3x

2x² + x² - 3x - 18 = 0

3x² - 3x - 18 = 0

∆ = (-3)² - 4 × 3 × (-18)

∆ = 9 + 216

∆ = 225

x = (3 ± √225)/(2 × 3)

x = (3 ± 15)/6

x' = -12/6 = -2

x" = 18/6 = 3

c) 2/(x² - 1) - x/(x - 1) = 2

2x² - 2 = -x² - x + 2

3x² + x - 4 = 0

∆ = 1² - 4 × 3 × (-4)

∆ = 1 + 48

∆ = 49

x = (-1 ± √49)/(2 × 3)

x = (-1 ± 7)/6

x' = -8/6 = -4/3

x" = 6/6 = 1

Espero Ter Ajudado !!