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vì \(\left(4x^2-4x+1\right)^{2022}\ge0\left(\forall x\right)\),\(\left(y^2-\dfrac{4}{5}y+\dfrac{4}{25}\right)^{2022}\ge0\left(\forall y\right)\),\(\left|x+y+z\right|\ge0\)
mà \(\left(4x^2-4x+1\right)^{2022}+\left(y^2+\dfrac{4}{5}y+\dfrac{4}{25}\right)^{2022}+\left|x+y-z\right|=0\)
=>\(\left\{{}\begin{matrix}4x^2-4x+1=0\\y^2+\dfrac{4}{5}y+\dfrac{4}{25}=0\\x+y-z=0\end{matrix}\right.\)
=>\(\left\{{}\begin{matrix}2x-1=0\\y+\dfrac{2}{5}=0\\x+y-z=0\end{matrix}\right.\)
<=>\(\left\{{}\begin{matrix}x=\dfrac{1}{2}\\y=\dfrac{-2}{5}\\\dfrac{1}{2}-\dfrac{2}{5}-z=0\end{matrix}\right.\)
<=>\(\left\{{}\begin{matrix}x=\dfrac{1}{2}\\y=\dfrac{-2}{5}\\z=\dfrac{1}{10}\end{matrix}\right.\)
KL: vậy \(\left\{{}\begin{matrix}x=\dfrac{1}{2}\\y=\dfrac{-2}{5}\\z=\dfrac{1}{10}\end{matrix}\right.\)
\(\dfrac{x+1}{x^2+2022}\) là số nguyên thì:
\(\left(x+1\right)⋮\left(x^2+2022\right)\\ \Rightarrow\left[\left(x+1\right)\left(x-1\right)\right]⋮\left(x^2+2022\right)\\ \Rightarrow\left(x^2+x-x-1\right)⋮\left(x^2+2022\right)\\ \Rightarrow\left(x^2-1\right)⋮\left(x^2+2022\right)\\ \Rightarrow\left(x^2+2022-2023\right)⋮\left(x^2+2022\right)\)
\(Mà.\left(x^2+2022\right)⋮\left(x^2+2022\right)\\ \Rightarrow2023⋮\left(x^2+2022\right)\\ \Rightarrow x^2+2022\inƯ\left(2023\right)\\ \Rightarrow x^2+2022\in\left\{-289;-119;-17;-7;-1;-2023;1;7;17;119;289;2023\right\}\)
Ta có: \(x^2+2022\ge0\Rightarrow x^2+2022=2023\Rightarrow x^2=1\Rightarrow x=\pm1\)
Vậy \(x=\pm1\) thì biểu thức trên là số nguyên
Sửa: \(Đk:x\ge0\)
\(C=1-\dfrac{1}{\sqrt{x}+2022}\ge1-\dfrac{1}{0+2022}=\dfrac{2021}{2022}\\ C_{min}=\dfrac{2021}{2022}\Leftrightarrow x=0\)
\(C=\dfrac{\sqrt{x}+2022}{\sqrt{x}+2022}-\dfrac{1}{\sqrt{x}+2022}=1-\dfrac{1}{\sqrt{x}+2022}\)
Do \(\sqrt{x}+2022\ge2022\Leftrightarrow\dfrac{1}{\sqrt{x}+2022}\le\dfrac{1}{2022}\Leftrightarrow-\dfrac{1}{\sqrt{x}+2022}\ge-\dfrac{1}{2022}\)
\(\Leftrightarrow C=1-\dfrac{1}{\sqrt{x}+2022}\ge1-\dfrac{1}{2022}=\dfrac{2011}{2022}\)
Dấu"=" xảy ra \(\Leftrightarrow x=0\)
tìm giá trị lớn nhất của P = \(\dfrac{|x-2022|-|x-2023|+|x-2024|+2022}{|x-2022|+|x-2023|+|x-2024|}\)
Lời giải:
Từ điều kiện đề bài suy ra:
$\frac{x}{y}=\frac{y}{z}=\frac{z}{x}$
$\Rightarrow (\frac{x}{y})^3=(\frac{y}{z})^3=(\frac{z}{x})^3=\frac{x}{y}.\frac{y}{z}.\frac{z}{x}=1$
$\Rightarrow \frac{x}{y}=\frac{y}{z}=\frac{z}{x}=1$
$\Rightarrow x=y=z$.
Do đó:
$\frac{(x+y+z)^{2022}}{x^{337}.y^{674}.z^{1011}}=\frac{(3x)^{2022}}{x^{337}.x^{674}.x^{1011}}=\frac{3^{2022}.x^{2022}}{x^{2022}}=3^{2022}$
Lời giải:
Từ điều kiện đề bài suy ra:
$\frac{x}{y}=\frac{y}{z}=\frac{z}{x}$
$\Rightarrow (\frac{x}{y})^3=(\frac{y}{z})^3=(\frac{z}{x})^3=\frac{x}{y}.\frac{y}{z}.\frac{z}{x}=1$
$\Rightarrow \frac{x}{y}=\frac{y}{z}=\frac{z}{x}=1$
$\Rightarrow x=y=z$.
Do đó:
$\frac{(x+y+z)^{2022}}{x^{337}.y^{674}.z^{1011}}=\frac{(3x)^{2022}}{x^{337}.x^{674}.x^{1011}}=\frac{3^{2022}.x^{2022}}{x^{2022}}=3^{2022}$
a)
`(2x-1)(x+2/3)=0`
\(< =>\left[{}\begin{matrix}2x-1=0\\x+\dfrac{2}{3}=0\end{matrix}\right.\\ < =>\left[{}\begin{matrix}x=\dfrac{1}{2}\\x=-\dfrac{2}{3}\end{matrix}\right.\)
b)
\(\dfrac{x+4}{2019}+\dfrac{x+3}{2020}=\dfrac{x+2}{2021}+\dfrac{x+1}{2022}\)
\(< =>\dfrac{x+4}{2019}+1+\dfrac{x+3}{2020}+1=\dfrac{x+2}{2021}+1+\dfrac{x+1}{2022}+1\)
\(< =>\dfrac{x+2023}{2019}+\dfrac{x+2023}{2020}=\dfrac{x+2023}{2021}+\dfrac{x+2023}{2022}\)
\(< =>\left(x+2023\right)\left(\dfrac{1}{2019}+\dfrac{1}{2020}-\dfrac{1}{2021}-\dfrac{1}{2022}\right)=0\)
\(< =>x+2023=0\left(\dfrac{1}{2019}+\dfrac{1}{2020}-\dfrac{1}{2021}-\dfrac{1}{2022}\ne0\right)\\ < =>x=-2023\)
\(\dfrac{2}{3}-\left|\dfrac{3}{4}\right|+\sqrt{\dfrac{25}{9}}-\left(\dfrac{2021}{2022}\right)^0=\dfrac{2}{3}-\dfrac{3}{4}+\dfrac{5}{3}-1=\dfrac{7}{12}\)
\(=\dfrac{2}{3}-\dfrac{3}{4}+\dfrac{5}{3}-1=\dfrac{7}{12}\)
\(\dfrac{x+23}{2021}+\dfrac{x+22}{2022}+\dfrac{x+21}{2023}+\dfrac{x+20}{2024}=-4\)
Vì \(\dfrac{x+23}{2021}+\dfrac{x+22}{2022}+\dfrac{x+21}{2023}+\dfrac{x+20}{2024}=-4\)
\(\Rightarrow\dfrac{x+23}{2021}+\dfrac{x+22}{2022}+\dfrac{x+21}{2023}+\dfrac{x+20}{2024}+4=0\)
\(\Rightarrow\left(\dfrac{x+23}{2021}+1\right)+\left(\dfrac{x+22}{2022}+1\right)+\left(\dfrac{x+21}{2023}+1\right)+\left(\dfrac{x+20}{2024}+1\right)=0\)
\(\Rightarrow\dfrac{x+2044}{2021}+\dfrac{x+2044}{2022}+\dfrac{x+2044}{2023}+\dfrac{x+2044}{2024}=0\)
\(\Rightarrow\left(x+2044\right)\left(\dfrac{1}{2021}+\dfrac{1}{2022}+\dfrac{1}{2023}+\dfrac{1}{2024}\right)=0\)
\(\Rightarrow x+2044=0\left(\dfrac{1}{2021}+\dfrac{1}{2022}+\dfrac{1}{2023}+\dfrac{1}{2024}\ne0\right)\)
\(\Rightarrow x=-2024\)
25x+2022−3x+∣−2022∣=2x+1011
�+20225−�+20223−�+20222=05x+2022−3x+2022−2x+2022=0
(15−13−12)(�+2022)=0(51−31−21)(x+2022)=0
(�+2022)=0(x+2022)=0