rozwiaz rownania i nierownosci wielomianowe

[woznyadam]

co do kilku nie jestem pewien wiec prosilbym o pomoc

\(\displaystyle{ f)x^{3}-2x^{2}+2=x}\)
\(\displaystyle{ g) x^{3}+8=0}\)
\(\displaystyle{ h) x^{4}-3x^{2}+2=0}\)
\(\displaystyle{ i) x^{4}-x{2}-2=0}\)

nierownosci
\(\displaystyle{ c) (x^{2}-5x+6)(x^{2}-1) qslant 0}\)
\(\displaystyle{ d) x^{3} qslant 9x}\)
\(\displaystyle{ e) a^{4} qslant a^{2}}\)
\(\displaystyle{ f) 9x^{3}+12x^{2}+4x qslant 0}\)
\(\displaystyle{ h) x^{4}+5x^{3}-x^{2}-5x}\)

[maise]

f)

\(\displaystyle{ x^3-2x^2-x+2=0\\
x(x^2-1)-2(x^2-1)=0\\
(x^2-1)(x-2)=0\\
(x^2-1)=0 (x-2)=0\\
x^2=1 \\
x=-1 x=1\\
x=2}\)



g)

\(\displaystyle{ x^3=8\\
x= \sqrt[3]{8} \\
x=2}\)



h)

\(\displaystyle{ x^2=t\\
t^2-3t+2=0\\
\Delta=(-3)^2-4*2*1\\
\Delta=1\\
\sqrt{\Delta} =1}\)
\(\displaystyle{ \begin{cases}
t _{1} = \frac{3-1}{2} \\
t _{2} = \frac{3+1}{2}
\end{cases}
\begin{cases}
t _{1} =1\\
t _{2}=2\\
\end{cases}}\)
\(\displaystyle{ \begin{cases}
x _{1} = \sqrt{t _{1}} \\
x _{2} = \sqrt{t _{2}}
\end{cases}

x=-1
x=1
x=- \sqrt{2}
x= \sqrt{2}}\)


i)

\(\displaystyle{ x^2=t\\
t^2-t-2=0\\
\Delta=(-1)^2-4*1*(-2)\\
\Delta=9\\
\sqrt{\Delta}=3\\
\begin{cases}
t _{1} = \frac{1-3}{2} \\
t _{2} = \frac{1+3}{2}
\end{cases}
\begin{cases}
t _{1} =-1\\
t _{2}=2\\
\end{cases}}\)

\(\displaystyle{ \begin{cases}
x _{1} = \sqrt{t _{1}} \\
x _{2} = \sqrt{t _{2}}
\end{cases}


x=- \sqrt{2}
x= \sqrt{2}}\)

[piasek101]

g)
\(\displaystyle{ x^3=8\\
x= \sqrt[3]{8} \\
x=2}\)

\(\displaystyle{ -}\)

[woznyadam]

dziekuej za pomoc, a mozna jeszcze prosic o pomoc przy nierownosciach?

[maise]

c)

\(\displaystyle{ (x^2-5x+6)(x^2-1) qslant 0\\
(x^2-5x+6) qslant 0 \ \ (x^2-1) qslant 0\\
\\
x^2-5x+6 qslant 0 \\
\Delta=9-5)^2-4 6 6\\
\Delta=25-24\\
\Delta=1\\
\sqrt{\Delta} =1\\
\begin{cases}
x_{1} qslant \frac{5-1}{2} \\
x_{2} qslant \frac{5+1}{2}\\
\end{cases}
\\
\begin{cases}
x_{1} qslant 2 \\
x_{2} qslant 3\\
\end{cases}
\\
2 qslant x qslant 3\\
\\
x^2-1 qslant 0\\
x^2 qslant 1\\
\begin{cases}
x_{1} qslant -1 \\
x_{2} qslant 1\\
\end{cases}
\\
-1 qslant x qslant 1
\\
\\
x \cup }\)


d)

\(\displaystyle{ x^3 \leqslant 9x\\
x^3-9x \leqslant 0\\
x(x^2-9) \leqslant 0\\
\Rightarrow x \leqslant 0 \ \vee \ (x^2-9) \leqslant 0\\
\\
x^2-9 \leqslant 0\\
x^2 \leqslant 9\\
-3 \leqslant x \leqslant 3
\\
x \in (- \infty ;3>}\)

[kolanko]

e)
\(\displaystyle{ a^{4}-a^{2} qslant 0}\)
\(\displaystyle{ a^{2}(a^{2}-1 qslant 0}\)
\(\displaystyle{ a^{2}(a-1)(a+1) qslant 0}\)
\(\displaystyle{ a (-\infty,-1) \cup {0} \cup (1,\infty)}\)










f)
\(\displaystyle{ x(9x^{2}+12x+4)\geqslant0}\)
\(\displaystyle{ \Delta = 144-144=0}\)
\(\displaystyle{ x_{0}=\frac{-b}{2a}=\frac{-12}{18}=\frac{-2}{3}}\)
\(\displaystyle{ x(9x^{2}+12x+4)\geqslant0}\)
\(\displaystyle{ x(3x+2)^{2} qslant 0}\)
\(\displaystyle{ x {\frac{-2}{3} \cup (0,\infty)}\)