Question

Rozwiąż równanie
log₂(x+1) +log₂(x-7) = 7
log₂(x-7) +log₂(x+5) = 0

Answer 1

log 2 [ x + 1] + log 2 [ x - 7 ] = 7

założenie: x + 1 > 0   ^   x - 7 > 0

log 2 [( x +1)*(x -7)] = 7

log 2 [ x^2 - 7x + x - 7 ] = log 2 [ 128 ]

x^2 - 6x - 7 = 128

x^2 - 6x - 135 = 0

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delta = (-6)^2 - 4*1*(-135) = 36 + 540 = 576

p(delty) = 24

x = [ 6 - 24]/2 = - 18/2 = - 9   <- nie spełnia założenia

lub

x = [ 6 + 24]/2 = 30/2 = 15

Odp. x = 15

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log 2 [ x - 7 ] + log 2 [ x + 5 ] = 0

założenia: x - 7 > 0  ^    x + 5  > 0

log 2 [ ( x -7)*( x + 5)] = 0

log 2 [ x^2 + 5x - 7x - 35 ] = log 2 [ 1 ]

x^2 - 2x - 35 = 1

x^2 - 2x - 36 = 0

delta = (-2)^2 - 4*1*( -36) = 4 + 144 = 148 = 4*37

p(delty) = 2 p(37)

x = [ 2 - 2 p(37)]/2 = 1 - p(37)  < 0  - odpada

lub

x = [ 2 + 2 p(37)]/2 = 1 + p(37) > 7

Odp. x = 1 + p(37)

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