a, \(Chof\left(x\right)=0\Rightarrow\left[{}\begin{matrix}x=3\\x=4\end{matrix}\right.\)

- Lập bảng xét dấu :

Vậy \(\left\{{}\begin{matrix}f\left(x\right)>0\Leftrightarrow x\in\left(3;4\right)\\f\left(x\right)< 0\Leftrightarrow x\in\left(-\infty;3\right)\cup\left(4;+\infty\right)\\f\left(x\right)=0\Leftrightarrow x\in\left\{3;4\right\}\end{matrix}\right.\)

b, \(f\left(x\right)=\left(x-1\right)\left(x+6\right)\)

( Làm tương tự câu a )

Nghiệm là 2.

Cho \(M\left(x\right)=0\)

hay \(x^2-3x+2=0\)

⇒    \(x^2-2x-x+2=0\)

     \(x.x-2x-x+2=0\)

  \(x.\left(x-2\right)-\left(x+2\right)=0\)

⇒     \(\left(x-1\right).\left(x-2\right)=0\)

⇒ \(x-1=0\) hoặc \(x-2=0\)

 * \(x-1=0\)        * \(x-2=0\)

    \(x\)      \(=0+1\)     \(x\)        \(=0+2\)

     \(x\)       \(=1\)          \(x\)        \(=2\)

Vậy \(x=1\) hoặc \(x=2\) là nghiệm của \(M\left(x\right)\)

\(a,2x\left(x^3-3\right)-2x^4=18\\ 2x^4-6x-2x^4=18\\ -6x=18\\ x=-3\)

\(b,9x\left(4-x\right)+\left(3x+1\right)^2=2\\ 36x-9x^2+9x^2+6x+1=2\\ 42x=2-1\\ 42x=1\\ x=\dfrac{1}{42}\)

\(a,\Leftrightarrow2x^4-3x-2x^4=18\Leftrightarrow-3x=18\Leftrightarrow x=-6\\ b,\Leftrightarrow36x-9x^2+9x^2+6x+1=2\\ \Leftrightarrow42x=1\Leftrightarrow x=\dfrac{1}{42}\)

\(x^3+3x^2+x+a=x^2\left(x-2\right)+5x\left(x-2\right)+11\left(x-2\right)+22+a=\left(x-2\right)\left(x^2+5x+11\right)+22+a⋮\left(x-2\right)\)

\(\Rightarrow22+a=0\Rightarrow a=-22\)

Ta có: \(\left\{{}\begin{matrix}3x-y=2m-1\\x+2y=3m+2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}6x-2y=4m-2\\x+2y=3m+2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}7x=7m\\y=3x-2m+1\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x=m\\y=m+1\end{matrix}\right.\)

Mặt khác: \(x^2+y^2=2m^2+2m+1=2\left(m^2+m+\dfrac{1}{2}\right)\)

                 \(=2\left(m^2+2\cdot m\cdot\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{4}\right)=2\left(m+\dfrac{1}{2}\right)^2+\dfrac{1}{2}\ge\dfrac{1}{2}\)

 Dấu bằng xảy ra \(\Leftrightarrow m+\dfrac{1}{2}=0\Leftrightarrow m=-\dfrac{1}{2}\)

  Vậy ...

1) \(\left(3x+2\right)^2-\left(x-6\right)^2=\left(3x+2-x+6\right)\left(3x+2+x-6\right)=\left(2x+8\right)\left(4x-4\right)=8\left(x+4\right)\left(x-1\right)\)

2) \(A=x^2+2y^2+2xy-2y+2021=\left(x^2+2xy+y^2\right)+\left(y^2-2y+1\right)+2020=\left(x+y\right)^2+\left(y-1\right)^2+2020\ge2020\)

\(minA=2020\Leftrightarrow\)\(\left\{{}\begin{matrix}x=-1\\y=1\end{matrix}\right.\)

\(\left(3x+2\right)^2-\left(x-6\right)^2=\left(3x+2-x+6\right)\left(3x+2+x-6\right)=\left(2x+8\right)\left(4x-4\right)=2.\left(x+4\right).4\left(x-1\right)=8\left(x-1\right)\left(x+4\right)\)

\(P\left(\dfrac{1}{2}\right)+Q\left(\dfrac{1}{2}\right)=-5.\left(\dfrac{1}{2}\right)^3+3\left(\dfrac{1}{2}\right)^2+\dfrac{2}{2}+5-5\left(\dfrac{1}{2}\right)^3+6\left(\dfrac{1}{2}\right)^2+\dfrac{2}{2}+5\)

\(P\left(\dfrac{1}{2}\right)+Q\left(\dfrac{1}{2}\right)=-\dfrac{5.1}{8}+\dfrac{3.1}{4}+6-\dfrac{5.1}{8}+\dfrac{6.1}{4}+6\)

\(P\left(\dfrac{1}{2}\right)+Q\left(\dfrac{1}{2}\right)=-\dfrac{5}{8}+\dfrac{3}{4}+6-\dfrac{5}{8}+\dfrac{3}{2}+6\)

\(P\left(\dfrac{1}{2}\right)+Q\left(\dfrac{1}{2}\right)=13\)

\(Q\left(x\right)-P\left(x\right)=6\)

\(-5x^3+6x^2+2x+5+5x^3-3x^2-2x-5=6\)

\(3x^2=6\)

\(x^2=2\)

\(=>x=\pm\sqrt{2}\)

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