Question

1) Determine as raízes da equação 2x² - 3x - 5 = 0

2) Achar as raízes das equações:

a) x² - x - 20 = 0
b) x² - 3x - 4 = 0
c) x² - 8x + 7 = 0
d) x² + x (x - 6) = 0
e) x (x + 3) = 5x
f) x (x - 3) - 2 (x - 3) = 6
g) (x + 5)² = 25
h) (x - 2)² = 4 - 9x

3) Dada a função y = 9x² - 8x - 1, determine os valores para os quais y = 0.

Answer 1

1) Determine as raízes da equação: 2x² - 3x - 5 = 0

a= 2 b= -3 c = -5

Δ = b² - 4ac

Δ = (-3)² -4.2.(-5)

Δ = 9 + 40 ∴ Δ = 49

x = (-b ⁺/₋√Δ)/2a

x' = [-(-3) + √49]/2.2 ∴ x' = [3 + 7]/4 ∴ x' = 10/4 ∴ x' = 5/2

x'' = [3 -7]/4 ∴ x'' = -4/4 ∴ x'' = -1

S = {5/2,-1}

2) Achar as raízes das equações:

a) x² -x - 20 = 0

a = 1 b= -1 c = -20

Δ = (-1)² -4.1.(-20) ∴ Δ = 1 + 80 ∴Δ = 81

x' = [- (-1) + √81]/2.1 ∴ x' = [1 + 9]/2 ∴ x' = 10/2 ∴ x' = 5

x'' = 1 - 9/2 ∴ x'' = -8/2 ∴ x'' = -4

S = {5, -4}

b) x² - 3x - 4 = 0

a = 1 b= -3 c = -4

Δ = (-3)² -4.1.(-4) ∴ Δ = 9 + 16 ∴ Δ = 25

x' = [-(-3) + √25]/2.1 ∴ x' = [3 + 5]/2 ∴x' = 8/2 ∴ x' = 4

x'' = [3-5]/2 ∴x'' = -2/2∴ x''= -1

S= {4,-1}

c) x² - 8x + 7 = 0

a = 1 b= -8 c = 7

Δ = (-8)² - 4.1.7 ∴ Δ = 64 - 28 ∴Δ = 36

x' = [(-8) + √36]/2.1 ∴ x' = [8 + 6]/2 ∴ x' = 14/2∴ x' = 7

x'' = [8-6]/2 ∴ x'' = -2/2 ∴ x' = -1

S = {7,-1}

d) x² + x (x-6) = 0

x² + x² - 6x = 0

2x² -6x = 0

a = 2 b =-6 c = 0

Δ= (-6)² - 4.1.0 ∴ Δ = 36 - 0 ∴Δ = 36

x' = [- (-6) + √36]/2.2 ∴ x' = [6+6]/4 ∴ x' = 12/4 ∴ x' = 3

x'' = [6-6]/4 ∴ x'' = 0/4 ∴x'' = 0

S= {3,0}

e) x (x+3) = 5x

x² + 3x - 5x = 0

x² - 2x = 0

a = 1 b= -2 c=0

Δ = (-2)² - 4.1.0 ∴ Δ = 4 - 0 ∴ Δ = 4

x' = [-(-2)+√4]/2.1 ∴ x' = [2+2]/2∴ x' = 4/2 ∴ x' = 2

x'' = [2-2]/2 ∴x'' = 0/2∴ x'' = 0

S= {2,0}

f) x (x-3) - 2 (x-3) = 6

x² -3x -2x +6 - 6 = 0

x² -5x = 0

a = 1 b = -5 c=0

Δ = (-5)² - 4.1.0 ∴ Δ = 25 - 0 ∴Δ = 25

x' = [-(-5) + √25]/2 ∴ x' = [5 + 5]/2∴x' = 10/2 ∴ x' = 5

x'' = [5-5]/2∴ x'' = 0/2 ∴x'' = 0

S = {5,0}

g) (x+5)² = 25

x² + 10x + 25 = 25

x² + 10x = 0

a = 1 b= 10 c= 0

Δ = 10² - 4.1.0 ∴ Δ = 100 - 0 ∴ Δ = 100

x' = [-10 + √100]/2.1 ∴ x' = [-10 + 10]/2∴ x' = 20/2 ∴x' = 10

x'' = [-10 -10]/2 ∴ x'' = -20/2 ∴ x'' = -10

S = {-10,0}

h) (x-2)² = 4 - 9x

x² - 4x + 4 - 4 + 9x = 0

x² + 5x = 0

a= 1 b= 5 c = 0

Δ = 5² - 4.1.0 ∴ Δ = 25

x' = [-5 + √25]/2.1 ∴x' = [-5+5]/2 ∴ x' = 0/2 ∴ x' = 0

x'' [-5-5]/2 ∴ x'' = -10/2 ∴ x'' = -5

S = {-5,0}

3) Dada a função y = 9x² -8x -1 , determine os valores para os quais y = 0.

9x² -8x -1 = 0

a = 9 b= -8 c= -1

Δ = (-8)² - 4.9. (-1) ∴ Δ = 64 + 36 ∴Δ = 100

x' = [-(-8) + √100]/2.9 ∴ x' = [8 + 10]/18∴x' = 18/18∴ x' = 1

x'' = [8 - 10]/18 ∴x'' = -2/18∴ x'' = -1/9

S= {1, -1/9}