Question

2)Determine as raízes reais das seguintes funções:
a) y=x²+5x+6

b) y=x²-7x+12

c) y=2x²+5x-3

d) y=-x² - x +30

e) y=x²-6x+9

f) y=x² - x +2

g) y= -x² + 12x – 20

h) y= x² -3x

i) y= 7x² + 1

j) y=3x²

l) y=x² -49

m) y=3x² -x +1​

Answer 1

Resposta:

a) y=x²+5x+6

x' = -2

x" = -3

b) y=x²-7x+12

x' = 4

x" = 3

c) y=2x²+5x-3

x' = 1/2

x" = -3

d) y=-x² - x +30

x' = -6

x" = 5

e) y=x²-6x+9

x' = 3

x" = 3

f) y=x² - x +2

Negativo!

g) y= -x² + 12x – 20

x' = 2

x" = 10

h) y= x² - 3x

x' = 0/2

x" = -3

i) y= 7x² + 1

Negativo!

j) y=3x²

x' = 0/6

x" = 0/6

l) y=x² -49

Negativo!

m) y=3x² -x +1​

Negativo!

Explicação passo-a-passo:

a) y=x²+5x+6

Δ = b²- 4.a.c

Δ = 5²- 4 × 1 × 6

Δ = 25 - 24

Δ = 1

x = -b±√Δ/2.a

x = -5 ± √1 / 2 × 1

x = -5 ± 1 / 2

x' = -5 + 1 / 2

x' = -4 /2

x' = -2

x" = -5 - 1 / 2

x" = - 6 / 2

x" = - 3

b) y=x²-7x+12

Δ = b²- 4.a.c

Δ = (-7)²- 4 × 1 × 12

Δ = 49 - 48

Δ = 1

x = -b±√Δ/2.a

x = - (-7) ± √1 / 2 × 1

x = 7 ± 1 / 2

x' = 7 + 1 / 2

x' = 8 / 2

x' = 4

x" = 7 - 1 / 2

x" = 6 / 2

x" = 3

c) y=2x²+5x-3

Δ = b²- 4.a.c

Δ = 5²- 4 × 2 × (-3)

Δ = 25 + 24

Δ = 49

x = -b±√Δ/2.a

x = -5 ± √7 / 2 × 2

x = -5 ± 7 / 4

x' = -5 + 7 / 4

x' = 2 / 4 (fatorando por 2)

x' = 1 / 2

x" = -5 - 7 / 4

x" = - 12 / 4

x" = - 3

d) y=-x² - x + 30

Δ = b²- 4.a.c

Δ = (-1)²- 4 × (-1) × 30

Δ = 1 + 120

Δ = 121

x = -b±√Δ/2.a

x = -(-1) ± √121 / 2 × (-1)

x = 1 ± 11 / - 2

x' = 1 + 11 / - 2

x' = 12 / - 2

x' = -6

x" = 1 - 11 / - 2

x" = - 10 / - 2

x" = 5

e) y = x² - 6x + 9 Δ = b²- 4.a.c

Δ = (-6)²- 4 × 1 × 9

Δ = 36 - 36

Δ = 0

x = -b±√Δ/2.a

x = -(-6) ± √0 / 2 × 1

x = 6 ± 0 / 2

x' = 6 + 0 / 2

x' = 6 /2

x' = 3

x" = 6 - 0 / 2

x" = 6 / 2

x" = 3

f)y = x² - x + 2 Δ = b²- 4.a.c

Δ = (-1)²- 4 × 1 × 2

Δ = 1 - 8

Δ = -7

g)y= -x² + 12x – 20 Δ = b²- 4.a.c

Δ = 12²- 4 × (-1) × (-20)

Δ = 144 - 80

Δ = 64

x = -b±√Δ/2.a

x = -12 ± √64 / 2 × (-1)

x = -12 ± 8 / -2

x' = -12 + 8 / -2

x' = -4 / -2

x' = 2

x" = -12 - 8 / -2

x" = - 20 / -2

x" = 10

h) y= x² -3x Δ = b²- 4.a.c

Δ = (-3)²- 4 × 1 × 0

Δ = 9 - 0

Δ = 9

x = -b±√Δ/2.a

x = -3 ± √9 / 2 × 1

x = -3 ± 3 / 2

x' = -3 + 3 / 2

x' = 0 / 2

x" = - 3 - 3 / 2

x" = - 6 / 2

x" = - 3

i) y= 7x² + 1 Δ = b²- 4.a.c

Δ = 0²- 4 × 7 × 1

Δ = 0 - 28

Δ = -28

j) y=3x² Δ = b²- 4.a.c

Δ = 0²- 4 × 3 × 0

Δ = 0 - 0

Δ = 0

x = -b±√Δ/2.a

x = -0 ± √0 / 2 × 3

x = -0 ± 0 / 6

x' = -0 + 0 / 6

x' = 0 / 6

x" = - 0 - 0 / 6

x" = 0 / 6

l) y=x² -49 Δ = b²- 4.a.c

Δ = 0²- 4 × 1 × (-49)

Δ = 0 - 196

Δ = -196

m) y = 3x² - x + 1

Δ = b²- 4.a.c

Δ = (-1)²- 4 × 3 × 1

Δ = 1 - 12

Δ = -11