Question

1) Calcule os quadrados a seguir (3 pts).
a) (x + 8)2
b) (x + 3y)2
c) (a – 7)2
2) Fatore os seguintes trinômios (3 pts).
a) x2 + 6x + 9
b) 4x2 – 4x + 1
c) x2 + 4x + 4
3) Resolva as equações fatorando os trinômios (6 pts).
a) x2 + 18x + 81 = 0
b) x2 – 2x + 1 = 0
c) 4x2 – 12x + 9 = 0
4) Resolva as equações pelo método de completar quadrados (APENAS POR ESTE MÉTODO) (18 pts).
a) x2 + 4x – 21 = 0
b) x2 – 12x – 13 = 0
c) x2 + 10x + 24 = 0
d) x2 – 4x + 3 = 0
e) x2 + 4x – 12 = 0
f) x2 – 2x – 3 = 0

Answer 1

Explicação passo-a-passo:

1)

a)

(x + 8)² = x² + 2.x.8 + 8²

(x + 8)² = x² + 16x + 64

b)

(x + 3y)² = x² + 2.x.3y + (3y)²

(x + 3y)² = x² + 6xy + 9y²

c)

(a - 7)² = a² - 2.a.7 + 7²

(a - 7)² = a² - 14a + 49

2)

a)

x² + 6x + 9 = x² + 3x + 3x + 9

x² + 6x + 9 = x.(x + 3) + 3.(x + 3)

x² + 6x + 9 = (x + 3).(x + 3)

x² + 6x + 9 = (x + 3)²

b)

4x² - 4x + 1 = 4x² - 2x - 2x + 1

4x² - 4x + 1 = 2x.(2x - 1) - 1.(2x - 1)

4x² - 4x + 1 = (2x - 1).(2x - 1)

4x² - 4x + 1 = (2x - 1)²

c)

x² + 4x + 4 = x² + 2x + 2x + 4

x² + 4x + 4 = x.(x + 2) + 2.(x + 2)

x² + 4x + 4 = (x + 2).(x + 2)

x² + 4x + 4 = (x + 2)²

3)

a)

x² + 18x + 81 = x² + 9x + 9x + 81

x² + 18x + 81 = x.(x + 9) + 9.(x + 9)

x² + 18x + 81 = (x + 9).(x + 9)

x² + 18x + 81 = (x + 9)²

(x + 9)² = 0

x + 9 = 0

x' = x" = -9

S = {-9}

b)

x² - 2x + 1 = x² - x - x + 1

x² - 2x + 1 = x.(x - 1) - 1.(x - 1)

x² - 2x + 1 = (x - 1).(x - 1)

x² - 2x + 1 = (x - 1)²

(x - 1)² = 0

x - 1 = 0

x' = x" = 1

S = {1}

c)

4x² - 12x + 9 = 4x² - 6x - 6x + 9

4x² - 12x + 9 = 2x.(x - 3) - 3.(2x - 3)

4x² - 12x + 9 = (2x - 3).(2x - 3)

4x² - 12x + 9 = (2x - 3)²

(2x - 3)² = 0

2x - 3 = 0

2x = 3

x' = x" = 3/2

S = {3/2}

4)

a)

x² + 4x - 21 = 0

x² - 3x + 7x - 21 = 0

x.(x - 3) + 7.(x - 3) = 0

(x - 3).(x + 7) = 0

• x - 3 = 0

x' = 3

• x + 7 = 0

x" = -7

S = {-7, 3}

b)

x² - 12x - 13 = 0

x² + x - 13x - 13 = 0

x.(x + 1) - 13.(x + 1) = 0

(x + 1).(x - 13) = 0

• x + 1 = 0

x' = -1

• x - 13 = 0

x" = 13

S = {-1, 13}

c)

x² + 10x - 24 = 0

x² - 2x + 12x - 24 = 0

x.(x - 2) + 12.(x - 2) = 0

(x - 2).(x + 12) = 0

• x - 2 = 0

x' = 2

• x + 12 = 0

x" = -12

S = {-12, 2}

d)

x² - 4x + 3 = 0

x² - x - 3x + 3 = 0

x.(x - 1) - 3.(x - 1) = 0

(x - 1).(x - 3) = 0

• x - 1 = 0

x' = 1

• x - 3 = 0

x" = 3

S = {1, 3}

e)

x² + 4x - 12 = 0

x² - 2x + 6x - 12 = 0

x.(x - 2) + 6.(x - 2) = 0

(x - 2).(x + 6) = 0

• x - 2 = 0

x' = 2

• x + 6 = 0

x" = -6

S = {-6, 2}

f)

x² - 2x - 3 = 0

x² + x - 3x - 3 = 0

x.(x + 1) - 3.(x + 1) = 0

(x + 1).(x - 3) = 0

• x + 1 = 0

x' = -1

• x - 3 = 0

x" = 3

S = {-1, 3}

Answer 2

Explicação passo-a-passo:

Álgebra.

1) Desenvolvimento de quadrados.

para desenvolver quadrados vamos seguir (a ± b)^2 = a^2±2ab+b^2

A)$\iff\sf{ (x + 8)^2~=~x^2+2*x*8+8^2}$

$\iff\sf{ (x + 8)^2~=~x^2+16x+64}$

B)

$\iff\sf{ (x + 3y)^2~=~x^2+2*x*(3y)+(3y)^2}$

$\iff\sf{ (x + 3y)^2~=~x^2+6xy+9y^2}$

C)

$\iff\sf{ (a - 7)^2~=~a^2-2*a*7+7^2}$

$\iff\sf{ (a - 7)^2~=~a^2+14a+49}$

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• Fatorizačão de trinomios :

2.

A)

$\iff \sf{ x^2+6x + 9 }$

$\iff \sf{ x^2+2*x*3 + 3^2 }$

$\iff \sf{ ( x + 3)^2 }$

B)

$\iff \sf{ 4x^2-4x + 1 }$

$\iff \sf{ (2x)^2 - 2*2x*1 + 1^2 }$

$\iff \sf{ (2x - 1)^2 }$

C)

$\iff \sf{ x^2 + 4x + 4 }$

$\iff \sf{ x^2 + 2*x*2 + 2^2 }$

$\iff \sf{ (x + 2)^2 }$

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3. Resolução de equações por Fatorizačão :

A)

$\iff \sf{ x^2 + 18x + 81~=~0 }$

$\iff \sf{ (x + 9)^2~=~0 }$

$\iff \sf{ x + 9 ~=~0 }$

$\iff \sf{ x_{1}~=~x_{2}~=~-9 }$

B)

$\iff \sf{ x^2 - 2x + 1 ~=~0 }$

$\iff \sf{ (x - 1)^2~=~0 }$

$\iff \sf{ x - 1 ~=~0 }$

$\iff \sf{ x_{1}~=~x_{2}~=~1 }$

C)

$\iff \sf{ 4x^2 - 12x + 9 ~=~0 }$

$\iff \sf{ (2x)^2 - 2*(2x)*3 + 3^2 }$

$\iff \sf{ (2x - 3)^2~=~0 }$

$\iff \sf{ 2x~=~3 }$

$\iff \sf{ x_{1}~=~x_{2}~=~\dfrac{3}{2} }$

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4. Resolução de equações por completamento de quadrados :

A)

$\iff \sf{ x^2 + 4x - 21 = 0 }$

$\iff \sf{ x^2 + 4x + 4 - 25 = 0 }$

$\iff \sf{ (x + 2)^2~=~25 }$

$\iff \sf{ x + 2 ~=~ \pm \sqrt{25}~=~\pm 5 }$

$\iff \sf{ x + 2 = 5~\vee ~x + 2 = -5 }$

$\iff \sf{ x = 5 - 2~\vee~x = -5- 2 }$

$\iff \red{ x_{1}~=~3~\vee~x_{2}~=~-7 }$

B)

$\iff \sf{ x^2 - 12x - 13 ~=~ 0 }$

$\iff \sf{ x^2 - 12x + 36 - 49 ~=~0}$

$\iff \sf{ (x - 6)^2~=~49 }$

$\iff \sf{ x - 6 ~=~\pm\sqrt{49}~=~\pm 7 }$

$\iff \sf{ x - 6 = 7 ~\vee~x - 6=-7 }$

$\iff \sf{ x_{1}=13~\vee~x_{2}=-1 }$

C)

$\iff \sf{ x^2+10x+24~=~0 }$

$\iff \sf{ x^2+10x + 25 - 1~=~0 }$

$\iff \sf{ (x + 5)^2~=~1 }$

$\iff \sf{ x + 5 = \pm\sqrt{1}~=~\pm 1 }$

$\iff \sf{ x + 5 = 1 ~\vee~x + 5 = -1 }$

$\iff \sf{ x_{1}~=~-4~\vee~x_{2}~=~-6 }$

D)

$\iff \sf{ x^2-4x + 3 = 0 }$

$\iff \sf{ x^2-4x + 4 - 1~=~0 }$

$\iff \sf{ (x - 2)^2~=~1 }$

$\iff \sf{ x - 2 ~=~\pm 1 }$

$\iff \sf{ x - 2 = 1 \vee ~ x - 2 = -1 }$

$\iff \sf{ x_{1}~=~3 \vee~x_{2}~=~1 }$

E)

$\iff \sf{ x^2 + 4x - 12 = 0 }$

$\iff \sf{ x^2 + 4x + 4 - 16 = 0 }$

$\iff \sf{ (x + 2)^2~=~16 }$

$\iff \sf{ x + 2 ~=~ \pm \sqrt{16}~=~4 }$

$\iff \sf{ x + 2 = 4 \vee~x + 2 = -4 }$

$\iff \sf{ x_{1}~=~2~\vee~x_{2}~=~-6 }$

F)

$\iff \sf{ x^2 - 2x - 3~=~0 }$

$\iff \sf{ x^2 - 2x + 1 - 4 ~=~0 }$

$\iff \sf{ (x - 1)^2~=~4 }$

$\iff \sf{ x - 1 = \pm\sqrt{4}~=~\pm 2 }$

$\iff \sf{ x - 1 = 2~\vee~x - 1 = -2 }$

$\iff \sf{ x_{1}~=~3~\vee~x_{2}~=~-1 }$

Espero ter ajudado bastante!)