Question

Resolver as equações:

A) (4-x) (2x+1) = 10

B) (x-2)^2 - 25 =0

C) 3(3-2x) + 10x = 3x(x+6)

D) (x-4)^2 + (x-5)(x+5) + 2x = 7 - 2x

E) (x-1)^2 + (x+5)(x-5) + (x-3)^2 + 3(1-x) = 4x - 30

Answer 1

a)
(4 - x).(2x + 1) = 10
8x + 4 - 2x² - x = 10
- 2x² + 7x + 4 - 10 = 0

- 2x² + 7x - 6 = 0
a = -2; b = 7; c = - 6

Δ = b² - 4ac
Δ = 7² - 4.(-2).(-6)
Δ = 49 + 8.(-6)
Δ = 49 - 48
Δ = 1

x = - b +/- √Δ = - 7 +/- 1
            2a          2(-2)

x = - 7 - 1 = - 8 / - 4 = 2
        - 4
x = - 7 + 1 = - 6 (:2) = 3
        - 4        -4  (:2)   2

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b)
(x - 2)² - 25 = 0
x² - 2.2x + 4 - 25 = 0

x² - 4x - 21 = 0
a = 1; b = - 4; c = - 21

Δ = b² - 4ac
Δ = (-4)² - 4.1.(-21)
Δ = 16 + 84
Δ = 100
√Δ = 10

x = - b +/- √Δ = - (-4) +/-√100 = 4 +/-10
          2a                 2.1               2
 
x = 4 + 10 = 14/2 = 7
         2

x = 4 - 10 = - 6/2 = - 3
         2
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c)
3(3 - 2x) + 10x = 3x(x + 6)
9 - 6x + 10x = 3x² + 18x
9 + 4x - 3x² - 18x = 0
- 3x² - 14x + 9 = 0
a = - 3; b = - 14; c = 9

Δ = b² - 4ac
Δ = (-14)² - 4.(-3).(9)
Δ = 196 + 12.(9)
Δ =196 + 108
Δ = 304
√Δ = √304 = √16.√19 = 4√19

x = - b +/- √Δ = - (-14) +/- √304 = 14 +/- 4√19
           2a                2.(-3)                - 6

x' = 14 + 4√19 = - 14 - 4 √19 = - 7 - 2√19 = - 7 - 2√19
          - 6              6    6            3   3               3

x" = 14 - 4√19 = - 14 + 4 √19 = - 7 + 2√19
            - 6            6     6                3
**************************************
304: 2
152 : 2
76: 2
38 : 2
19: 19
1

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d)
(x - 4)² + (x - 5).(x + 5) + 2x = 7 - 2x
x² - 2.4x + 16 + x² + 5x - 5x - 25 + 2x = 7 - 2x
x² - 8x + 16 + x² - 25 + 2x - 7 + 2x = 0
x² + x² - 8x + 2x + 2x + 16 - 25 - 7 = 0

2x² - 4x - 16 = 0
a = 2; b = -4; c = - 16

Δ = b² - 4ac
Δ = (-4)² - 4.2.(-16)
Δ = 16 - 8.(-16)
Δ = 16 + 128
Δ = 144
√Δ = 12

x = - b +/- √Δ = - (-4) +/- √144 = 4 +/- 12
           2a                2.2                4

x = 4 + 12  = 16/4 = 4
          4
x = 4 - 12 = - 8/4 = - 2
         4
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e)
(x - 1)² + (x + 5).(x - 5) + (x - 3)² + 3(1 - x) = 4x - 30
x² - 2x + 1 + x² - 5x + 5x - 25 + x² - 2.3x + 9 + 3 - 3x = 4x - 30
x² - 2x + 1 + x² - 25 + x² - 6x + 12 - 3x - 4x + 30 = 0
x² + x² + x² - 2x - 6x - 3x - 4x  + 1 - 25 + 12 + 30 = 0

3x² - 15x + 18 = 0 (:3)
x² - 5x + 6 = 0
a = 1; b = - 5; c = 6

Δ = b² - 4ac
Δ = (-5)² - 4.1.6
Δ = 25 - 24
Δ = 1

x = 5 + 1 = 6/2 = 3
        2

x = 5 - 1 = 4/2 = 2
        2