Wyznacz postać ogólną trójmianu kwadratowego, znając jego pierwiastki oraz współczynnik a:
a)x1=3 x2=3/5 a=2
b)x1=-4 x2=1/3 a=-1/2
c)x1=pierwiastek z 3 x2=-2 a=-2/3
d)x1=3pierwiastek z 2 x2=1+ pierwiastekz2 a=-3
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Korzystamy z postaci iloczynowej trójmianu kwadratowego:
y = a*9x -x1)*(x - x2)
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a)
x1 = 3, x2 = 3/5 , a = 2
zatem
y = 2*(x -3)*(x - 3/5) = 2*(x^2 - (3/5)x - 3x + 9/5) = 2*(x^2 - 3,6 x + 2,25) =
= 2 x^2 - 7,2 x + 4,5
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b)
x1 = - 4, x2 = 1/3, a = - 1/2
y = (-1/2)*(x + 4)*( x - 1/3)
y = - (1/2)*(x^2 - (1/3)x + 4x - 4/3)
y = (-1/2) x^2 + (1/6)x - 2x + 4/6
y = (-1/2) x^2 - (11/6)x + 2/3
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c)
x1 = p(3), x2 = - 2, a = - 2/3
y = (-2/3)*(x - p(3)*(x + 2)
y = (-2/3)*(x^2 + 2x - p(3)x - 2 p(3))
y = (-2/3)*(x^2 + (2 - p(3))x -2 p(3))
y = (-2/3) x^2 + ( (2 p(3)/3 - 4/3)x + (4/3) p(3)
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d)
x1 = 3 p(2), x2 = 1 + p(2), a = - 3
zatem
y = - 3*( x - 3 p(2)) *(x - 1 - p(2))
y = -3*(x^2 -(1 + p(2))x - 3p(2) x + 3 + 3 p(2))
y = -3*(x^2 - (1 + 4 p(2))x + 3 p(2) + 3)
y = - 3 x62 + (3 + 12 p(2)) x - 9 p(2) - 9
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