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2 tháng 9 2017

câu b đk x>= -1/4

\(x+\sqrt{x+\dfrac{1}{2}+\sqrt{x+\dfrac{1}{4}}}=2\)

\(x+\sqrt{\left(\sqrt{x+\dfrac{1}{4}}+\dfrac{1}{2}\right)^2}=2\)

\(\left(\sqrt{x+\dfrac{1}{4}}+\dfrac{1}{2}\right)^2=2\)

\(x+\dfrac{1}{4}=\left(\sqrt{2}-\dfrac{1}{2}\right)^2\)

\(x=\left(\sqrt{2}-\dfrac{1}{2}\right)^2-\dfrac{1}{4}\)

\(x=\left(\sqrt{2}-\dfrac{1}{2}-\dfrac{1}{2}\right)\left(\sqrt{2}-\dfrac{1}{2}+\dfrac{1}{2}\right)\)

\(x=\sqrt{2}\left(\sqrt{2}-1\right)=2-\sqrt{2}\)

3 tháng 9 2017

bạn ghi cai gì vậy hả. Mình chẳng hiểu gì hết ý

8 tháng 1 2018

giải bài nào hộ mk cx được ko cần lm hết đâu :) :) :)

11 tháng 9 2016

\(A=\sqrt{\left(x-2\right)\left(x-1\right)x\left(x+1\right)+5}\)

\(=\sqrt{\left(x^2-x-2\right)\left(x^2-x\right)+5}\)

Đặt \(t=x^2-x\) ta đc:

\(A=\sqrt{\left(t-2\right)t+5}=\sqrt{t^2-2t+5}\)

\(=\sqrt{\left(t-1\right)^2+4}\ge\sqrt{4}=2\)

Dấu = khi \(t=1\Leftrightarrow x^2-x=1\Leftrightarrow x=\pm\frac{1}{2}+\frac{\sqrt{5}}{2}\)

Vậy....

b)\(B=\sqrt{x^2-4x+4}+\sqrt{x^2+6x+9}\)

\(=\sqrt{\left(x-2\right)^2}+\sqrt{\left(x+3\right)^2}\)

\(=\left|x-2\right|+\left|x+3\right|\)

Áp dụng Bđt \(\left|a\right|+\left|b\right|\ge\left|a+b\right|\) ta có:

\(\left|x-2\right|+\left|x+3\right|=\left|x-2\right|+\left|-x-3\right|\ge\left|x-2+\left(-x\right)-3\right|=5\)

Dấu = khi \(\left(x-2\right)\left(x+3\right)\ge0\)\(\Rightarrow-3\le x\le2\)

\(\Rightarrow\hept{\begin{cases}-3\le x\le2\\\left(x+3\right)\left(x-2\right)=0\end{cases}}\Leftrightarrow\hept{\begin{cases}x=-3\\x=2\end{cases}}\)

Vậy....

14 tháng 9 2017

1) ĐK: \(x\ge-2012\)

Đặt \(\sqrt{x+2012}=t\left(t\ge0\right)\Rightarrow x=t^2-2012\)

Ta có hệ \(\hept{\begin{cases}x^2+t=2012\\-x+t^2=2012\end{cases}}\)

\(\Rightarrow x^2+t-t^2+x=0\Rightarrow\left(x+t\right)\left(x-t+1\right)=0\)

Với \(x+t=0\Leftrightarrow\sqrt{x+2012}=x\Rightarrow x^2-x-2012=0\Rightarrow x=\frac{\sqrt{8049}+1}{2}\)

Với \(x-t+1=0\Leftrightarrow\sqrt{x+2012}=x+1\Rightarrow x^2+x-2011=0\Rightarrow x=\frac{\sqrt{8045}-1}{2}\)

2) ĐK \(\orbr{\begin{cases}x< -\frac{1}{3}\\x>1\end{cases}}\)

Đặt \(\sqrt{\frac{3x+1}{x-1}}=t\), phương trình trở thành \(4t+\frac{1}{t}=4\Rightarrow\frac{4t^2-4t+1}{t}=0\Rightarrow t=\frac{1}{2}\)

Khi đó ta có \(\sqrt{\frac{3x+1}{x-1}}=\frac{1}{2}\Rightarrow\frac{3x+1}{x-1}=\frac{1}{4}\Rightarrow11x+5=0\)

\(\Rightarrow x=-\frac{5}{11}\left(tm\right)\)

c) TH1: \(x\le-1\), phương trình trở thành \(\left(x-3\right)\left(x+1\right)-4\sqrt{\left(x-3\right)\left(x+1\right)}+3=0\)

Đặt \(\sqrt{\left(x-3\right)\left(x+1\right)}=t\left(t\ge0\right)\) thì \(t^2-4t+3=0\Rightarrow\orbr{\begin{cases}t=1\\t=3\end{cases}}\)

Với \(t=1\Rightarrow\left(x-3\right)\left(x+1\right)=1\Rightarrow x^2-2x-4=0\Rightarrow\orbr{\begin{cases}x=1+\sqrt{5}\left(l\right)\\x=1-\sqrt{5}\left(tm\right)\end{cases}}\)

Với \(t=3\Rightarrow\left(x-3\right)\left(x+1\right)=9\Rightarrow x^2-2x-12=0\Rightarrow\orbr{\begin{cases}x=1+\sqrt{13}\left(l\right)\\x=1-\sqrt{13}\left(tm\right)\end{cases}}\)

Với \(x>3\), phương trình trở thành \(\left(x-3\right)\left(x+1\right)+4\sqrt{\left(x-3\right)\left(x+1\right)}+3=0\)

Đặt \(\sqrt{\left(x-3\right)\left(x+1\right)}=t\left(t\ge0\right)\) thì \(t^2+4t+3=0\Rightarrow\orbr{\begin{cases}t=-1\\t=-3\end{cases}\left(l\right)}\)

Vậy pt có 2 nghiệm \(x=1-\sqrt{5}\) hoặc \(x=1-\sqrt{13}\)

23 tháng 6 2021

a) Áp dụng bđt AM-GM có:

\(\sqrt[3]{\left(9-x\right).8.8}\le\dfrac{9-x+8+8}{3}=\dfrac{25-x}{3}\)\(\Leftrightarrow\sqrt[3]{9-x}\le\dfrac{25-x}{12}\)

\(\sqrt[3]{\left(7+x\right).8.8}\le\dfrac{7+x+8+8}{3}=\dfrac{23+x}{3}\)\(\Leftrightarrow\sqrt[3]{7+x}\le\dfrac{23+x}{12}\)

Cộng vế với vế \(\Rightarrow\sqrt[3]{9-x}+\sqrt[3]{7+x}\le4\)

Dấu "=" xảy ra khi \(\left\{{}\begin{matrix}9-x=8\\7+x=8\end{matrix}\right.\)\(\Rightarrow x=1\)

Vậy...

b)Đk:\(x\ge2\)

Pt \(\Leftrightarrow\left(x-1\right)^2.\left(x^2-4\right)=\left(x-2\right)^2.\left(x^2-1\right)\)

\(\Leftrightarrow\left(x-1\right)^2\left(x-2\right)\left(x+2\right)=\left(x-2\right)^2\left(x+1\right)\left(x-1\right)\)

Do \(x\ge2\Rightarrow x-1>0\)

Chia cả hai vế của pt cho x-1 ta được:

\(\left(x-1\right)\left(x-2\right)\left(x+2\right)=\left(x-2\right)^2\left(x+1\right)\)

\(\Leftrightarrow\left(x-2\right)\left[\left(x-1\right)\left(x+2\right)-\left(x-2\right)\left(x-1\right)\right]=0\)

\(\Leftrightarrow\left(x-2\right)\left[x^2+x-2-x^2+3x-2\right]=0\)

\(\Leftrightarrow\left(x-2\right)\left(4x-4\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2\left(tm\right)\\x=1\left(ktm\right)\end{matrix}\right.\)

Vậy S={2}

c)Đk:\(\left\{{}\begin{matrix}9-x^2\ge0\\x^2-1\ge0\\x-3\ge0\end{matrix}\right.\)\(\Leftrightarrow\left\{{}\begin{matrix}-3\le x\le3\\\left[{}\begin{matrix}x\ge1\\x\le-1\end{matrix}\right.\\x\ge3\end{matrix}\right.\)\(\Rightarrow x=3\)

Thay x=3 vào pt thấy thỏa mãn

Vậy S={3}

23 tháng 6 2021

a) Quên mất, ko áp dụng đc AM-GM, xin lỗi

Pt \(\Leftrightarrow\sqrt[3]{9-x}-2=2-\sqrt[3]{7+x}\)

\(\Leftrightarrow\dfrac{9-x-8}{\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4}=\dfrac{8-\left(7-x\right)}{4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}}\)

\(\Leftrightarrow\dfrac{1-x}{\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4}=\dfrac{1-x}{4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}}\)

\(\Leftrightarrow\left[{}\begin{matrix}x-1=0\\\dfrac{1}{\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4}=\dfrac{1}{4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}}\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x=1\\\sqrt[3]{\left(9-x\right)^2}+2\sqrt[3]{9-x}+4=4+2\sqrt[3]{7+x}+\sqrt[3]{\left(7+x\right)^2}\left(1\right)\end{matrix}\right.\)

Từ (1) \(\Leftrightarrow\sqrt[3]{\left(9-x\right)^2}-\sqrt[3]{\left(7+x\right)^2}+2\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)=0\)

\(\Leftrightarrow\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)\left(\sqrt[3]{9-x}+\sqrt[3]{7+x}\right)+2\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)=0\)

\(\Leftrightarrow\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right).4+2\left(\sqrt[3]{9-x}-\sqrt[3]{7+x}\right)=0\)

\(\Leftrightarrow\sqrt[3]{9-x}-\sqrt[3]{7+x}=0\)

\(\Leftrightarrow\sqrt[3]{9-x}=\sqrt[3]{7+x}\)\(\Leftrightarrow9-x=7+x\)

\(\Leftrightarrow x=1\)

Vậy S={1}