Question

Rozwiąż równania:

A)x³-7x+6=0

B) 2x(do potęgi 4)-x³-3x²=0

C) 4x³+4x²-25x-25=0

D) 7x³+2x²-21x-6=0

Answer 1

a)

x^3 - 7 x + 6 = 0

x^3 - x - 6 x + 6 = 0

x*( x^2 - 1) - 6*( x - 1) = 0

x*( x - 1)*(x + 1) - 6*( x - 1) = 0

(x - 1)*[ x*( x + 1) - 6 ] = 0

( x - 1)*( x^2 + x - 6) = 0

x - 1 = 0  lub  x^2 + x - 6 = 0

x = 1

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( x - 2)*(x + 3) = 0

x - 2 = 0  lub  x + 3 = 0

x = 2  lub  x = - 3

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Odp.  x = - 3  lub  x = 1   lub  x = 2

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b)

2 x^4 - x^3  - 3 x^2 = 0

x^2 *( 2 x^2 - x - 3 ) = 0

x = 0  lub   2 x^2 - x - 2 = 0

delta = (-1)^2 - 4*2*(-2) = 1 + 16 = 17

x = [ 1 - p(17)]/4          lub  x = [ 1 + p(17)]/4

Odp.  x =  [ 1 - p(17)]/4   lub  x = 0  lub   x = [ 1 + p(17)]/4

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p(17)  - pierwiastek kwadratowy z  17

 

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c)

4 x^3 + 4 x^2 - 25 x - 25 = 0

4x^2 *( x + 1)  - 25*( x + 1) = 0

( x + 1)*( 4 x^2 - 25) = 0

( x + 1)*( 2 x - 5)*( 2 x + 5) = 0

x + 1 = 0  lub  2 x - 5 = 0    lub    2 x + 5 = 0

x = - 1     lub     x = 2,5      lub       x =  - 2,5

Odp.  x = - 2,5  lub  x = - 1    lub  x = 2,5

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d)

7 x^3 + 2 x^2 - 21 x - 6 = 0

x^2 *( 7 x + 2) - 3 *( 7 x + 2) = 0

( 7 x + 2)*( x^2 - 3 ) = 0

( 7 x + 2)*( x - p(3))*( x + p(3)) = 0

7 x + 2 = 0   lub  x - p(3) = 0   lub   x + p(3) = 0

x = - 2/7      lub      x = p(3)    lub   x = - p(3)

Odp.

x = - p(3)   lub  x = - 2/7    lub  x = p(3)

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p(3) - pierwiastek kwadratowy z 3