Determine o conjunto solução de cada uma das seguintes equações do 2º grau, sendo U=R.
a) x² - 4/5 x = 1/5
b) x + x²4/5 = 2
c) x²/2 - x+12/3 = 2x
d) x(x+1)/4 - x-5/12 = 5(2x-1)/6
a) x² - 4/5 x = 1/5
b) x + x²4/5 = 2
c) x²/2 - x+12/3 = 2x
d) x(x+1)/4 - x-5/12 = 5(2x-1)/6
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(x + 5).(x - 6) = 51 - x
x² - 6x + 5x - 30 = 51 - x
x² - x - 30 = 51 - x
x² = 51 + 30 + x - x
x² = 81
x = √81
x = +/-9
R.: x = 9 e x = -9
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b)
x² + 3x(x - 12) = 0
x² + 3x² - 36x = 0
4x² - 36x = 0
x(4x - 36) = 0
x = 0
ou
4x - 36 = 0
4x = 36
x = 36/4
x = 9
R. x = 0 e x = 9
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c)
(x - 5)² = 25 - 9x
x² - 2.5x + 25 = 25 - 9x
x² - 10 + 25 = 25 - 9x
x² - 10 + 9x = 0
a = 1; b = - 10; c = 9
Δ = b² - 4ac
Δ = (-10)² - 4.1.9
Δ = 100 - 36
Δ = 64
x = - b +/- √Δ = - ( - 10) +/- √64
2a 2.1
x = 10 + 8 = 18/2 = 9
2
x = 10 - 8 = 2/2 = 1
2
R.: x = 9 e x = 1
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d)
2x(x + 1) - x(x + 5) = 3(12 - x)
2x² + 2x - x² - 5x = 36 - 3x
x² - 3x = 36 - 3x
x² = 36
x = √36
x = +/- 6
R.: x = 6 e x = - 6
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e)
x + 2(x - 16) + (x + 7)² = 89
x + 2x - 32 + x² + 2.7x + 49 - 89 = 0
3x - 32 + x² + 14x - 40 = 0
x² + 17x - 72 = 0
a = 1; b = 17; c = - 72
Δ = b² - 4ac
Δ = 17² - 4.1.( - 72)
Δ = 289 - 4.( -72)
Δ = 289 - 288
Δ = 1
x = - b +/- √Δ = - 17 +/- √1
2a 2
x = - 17 + 1 = - 16/2 = - 8
2
x = - 17 - 1 = - 18/2 = - 9
2
R.: x = - 8 e x = - 9
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f)
(x - 4)² + 5x (x - 1) = 16
x² - 2.4x + 16 + 5x² - 5x - 16 = 0
x² - 8x + 16 + 5x² - 5x - 16 = 0
6x² - 13x = 0
x(6x - 13) = 0
x = 0
6x - 13 = 0
6x = 13
x = 13/6
R.: x = 0 e x = 13/6