rozwiąż równania

[adwrer]

Mógłby ktoś??

a) x^3-2x^2=5x-6

b) x^3-7x^2+15x=9

c) 3x^3-3x^2+4x-4=0

d) x^3+2x-16x-32=0

e) (x^5-x^3+x^2-1)(x^2-4)=0

f) 24x^4+16x^3+3x+2=0

[irena]

a)
\(x^3-2x^2=5x-6\\x^3-2x^2-5x+6=0\\x^3-x^2-x^2+x-6x+6=0\\x^2(x-1)-x(x-1)-6(x-1)=0\\(x-1)(x^2-x-6)=0\\x-1=0\ \vee \ x^2-x-6=0\\\Delta=1+24=25\\x_1=\frac{1-5}{2}=-2\ \vee \ x_2=\frac{1+5}{2}=3\\x=1\ \vee \ x=3\ \vee \ x=-2\)

[irena]

b)
\(x^3-7x^2+15x=9\\x^3-7x^2+15x-9=0\\x^3-x^2-6x^2+6x+9x-9=0\\x^2(x-1)-6x(x-1)+9(x-1)=0\\(x-1)(x^2-6x+9)=0\\(x-1)(x-3)^2=0\\x=1\ \vee \ x=3\)

[irena]

c)
\(3x^3-3x^2+4x-4=0\\3x^2(x-1)+4(x-1)=0\\(x-1)(3x^2+4)=0\\3x^2+4>0\\x=1\)

[irena]

d)
\(x^3+2x^2-16x-32=0\\x^2(x+2)-16(x+2)=0\\(x+2)(x^2-16)=0\\(x+2)(x+4)(x-4)=0\\x=-2\ \vee \ x=-4\ \vee \ x=4\)

[irena]

e)
\((x^5-x^3+x^2-1)(x^2-4)=0\\(x^3(x^2-1)+x^2-1)(x-2)(x+2)=0\\(x^2-1)(x^3+1)(x-2)(x+2)=0\\(x-1)(x+1)(x+1)(x^2-x+1)(x-2)(x+2)=0\\(x-1)(x+1)^2(x-2)(x+2)(x^2-x+1)=0\\x^2-x+1>0\\x=1\ \vee \ x=-1\ \vee \ x=2\ \vee \ x=-2\)

[irena]

f)
\(24x^4+16x^3+3x+2=0\\8x^3(3x+2)+3x+2=0\\(3x+2)(8x^3+1)=0\\(3x+2)(2x+1)(4x^2-2x+1)=0\\4x^2-2x+1>0\\x=-\frac{2}{3}\ \vee \ x=-\frac{1}{2}\)