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a)3(x-1)(x+2)-2x(x-1)(x+3)=0
3(x²+x -2) -2x(x²+2x -3) = 0
3x² +3x -6 -2x³ - 4x² +6x = 0
-2x³ -x² +9x -6 = 0
W(1) = -2*1³ -1² + 9*1 - 6 = -2 -1 +9 -6 = 9 -9 =0
czyli wielomian dzieli się bez reszty przez (x-1)
(-2x³ -x² +9x -6 ) : ( x-1) = -2x²-3x +6
+2x³-2x²
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= -3x² +9x -6
+3x²-3x
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= 6x -6
-6x +6
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= =
(-2x³ -x² +9x -6 ) = ( x-1) ( -2x²-3x +6)
z drugiego nawisu obliczam pierwiastki
Δ = (-3)² -4*(-2)*6 = 9 +54 = 63
√Δ = √63 = √9*√7 = 3√7
x1 = (3 -3√7): 2*(-2) = -3/4 +3/4√7
x2 = (3 +3√7): 2*(-2) = -3/4 -3/4√7
x = 1, lub x = -3/4 + 3/4*√7, lub x = -3/4 - 3/4*√7
b))-2x²+x+1=0
Δ = 1² -4*(-2)*1 = 1 +8 = 9
√Δ = √9 = 3
x1 = (-1 -3): 2*(-2) = (-4): (-4) = 1
x2 = (-1 +3): 2*(-2) = 2 : (-4) = -1/2
c)9x²+3x+2 = 0
Δ = 3² -4*9*2 = 9 - 72 = - 63
Δ < 0, więc brak jest pierwiastków
d)3x²-5x=x²+2x-3
3x²-5x - x²- 2x +3 = 0
2x² -7x +3 = 0
Δ = (-7)² -4*2*3 = 49 - 24 = 25
√Δ = √25 = 5
x1 = (7 -5):2*2 = 2 : 4 = 1/2
x2 = (7 +5): 2*2 = 12 :4 = 3