Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(x\ge-3\)
\(x^4\left(\sqrt{x+3}-2\right)+2016\left(x-1\right)=0\)
\(\Leftrightarrow\dfrac{x^4\left(x-1\right)}{\sqrt{x+3}+2}+2016\left(x-1\right)=0\)
\(\Leftrightarrow\left(x-1\right)\left(\dfrac{x^4}{\sqrt{x+3}+2}+2016\right)=0\)
\(\Leftrightarrow x-1=0\) (do \(\dfrac{x^4}{\sqrt{x+3}+2}+2016>0\) \(\forall x\ge-3\) )
\(\Rightarrow x=1\)
Vậy pt có nghiệm duy nhất \(x=1\)
Giải phương trình:
a)\(\sqrt{x^2+2x\sqrt{3}+3}=\sqrt{3}+x\)
b)\(\sqrt{x-3+2\sqrt{x-4}}=2\sqrt{x-4}+1\)
a)Pt\(\Leftrightarrow\sqrt{\left(x+\sqrt{3}\right)^2}=x+\sqrt{3}\)
\(\Leftrightarrow\left|x+\sqrt{3}\right|=x+\sqrt{3}\)
\(\Leftrightarrow x+\sqrt{3}\ge0\)\(\Leftrightarrow x\ge-\sqrt{3}\)
Vậy...
b)Đk:\(x\ge4\)
Pt\(\Leftrightarrow\sqrt{\left(x-4\right)+2\sqrt{x-4}+1}=2\sqrt{x-4}+1\)
\(\Leftrightarrow\sqrt{\left(\sqrt{x-4}+1\right)^2}=1+2\sqrt{x-4}\)
\(\Leftrightarrow\sqrt{x-4}+1=2\sqrt{x-4}+1\)
\(\Leftrightarrow\sqrt{x-4}=0\)
\(\Leftrightarrow x=4\) (tm)
Vậy...
a) Ta có: \(\sqrt{x^2+2x\sqrt{3}+3}=x+\sqrt{3}\)
\(\Leftrightarrow\left|x+\sqrt{3}\right|=x+\sqrt{3}\)
\(\Leftrightarrow\left[{}\begin{matrix}x+\sqrt{3}=x+\sqrt{3}\left(x\ge-\sqrt{3}\right)\\x+\sqrt{3}=-x-\sqrt{3}\left(x< -\sqrt{3}\right)\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}x\ge-\sqrt{3}\\x=-\sqrt{3}\left(loại\right)\end{matrix}\right.\Leftrightarrow x\ge-\sqrt{3}\)
b, \(đk:x\ge2\)
Xét x=2 thay vào pt thấy không thỏa mãn => x>2 hay 27x-54>0
\(x^3-11x+36x-18=4\sqrt[4]{27x-54}\)
\(\Leftrightarrow27x^3-297x^2+972x-486=4\sqrt[4]{\left(27x-54\right).81.81.81}\le189+27x\) (cosi với 4 số dương, dấu = xảy ra khi x=5)
\(\Leftrightarrow x^3-11x^2+35x-25\le0\)
\(\Leftrightarrow\left(x-1\right)\left(x-5\right)^2\le0\) (*)
Có \(\left\{{}\begin{matrix}x>2\\\left(x-5\right)^2\ge0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x-1>0\\\left(x-5\right)^2\ge0\end{matrix}\right.\)\(\Rightarrow\left(x-1\right)\left(x-5\right)^2\ge0\) (2*)
Từ (*) và (2*) ,dấu = xra khi x=5 (thỏa mãn)
Vây pt có nghiệm duy nhất x=5
c,Có \(6\sqrt[3]{4x^3+x}=16x^4+5>0\)
\(\Leftrightarrow4x^3+x>0\)
Có: \(16x^4+5=6\sqrt[3]{4x^3+x}\le2\left(4x^3+x+2\right)\) (theo cosi với 3 số dương,dấu = xảy ra khi \(x=\dfrac{1}{2}\))
\(\Leftrightarrow16x^4-8x^3-2x+1\le0\)
\(\Leftrightarrow\left(2x-1\right)^2\left(4x^2+2x+1\right)\le0\) (*)
(tương tự câu b) Dấu = xảy ra khi \(x=\dfrac{1}{2}\)(thỏa mãn)
Vậy....
d) Đk: \(x\ge\dfrac{3}{4}\)
Áp dụng bđt cosi:
\(\sqrt{2x-1}\le\dfrac{2x-1+1}{2}=x\)
\(\Rightarrow\dfrac{1}{\sqrt{2x-1}}\ge\dfrac{1}{x}\) (*)
\(\sqrt[4]{4x-3}\le\dfrac{4x-3+1+1+1}{4}=x\)
\(\dfrac{\Rightarrow1}{\sqrt[4]{4x-3}}\ge\dfrac{1}{x}\) (2*)
Từ (*) và (2*) \(\Rightarrow\dfrac{1}{\sqrt{2x-1}}+\dfrac{1}{\sqrt[4]{4x-3}}\ge\dfrac{2}{x}\)
Dấu = xảy ra khi x=1 (tm)
\(a,\left(đk:x\ge0\right)\)
\(x=0\Rightarrow\sqrt{0+3}+0=0\left(vô-nghiệm\right)\)
\(x>0\)
\(\)\(\sqrt{x+3}+\dfrac{4x}{\sqrt{x+3}}=4\sqrt{x}\Leftrightarrow\dfrac{\sqrt{x+3}}{\sqrt{x}}+\dfrac{4\sqrt{x}}{\sqrt{x+3}}=4\)
\(VT\ge2\sqrt{\dfrac{\sqrt{x+3}}{\sqrt{x}}.\dfrac{4\sqrt{x}}{\sqrt{x+3}}}=4\)
\(dấu"="xảy-ra\Leftrightarrow\dfrac{\sqrt{x+3}}{\sqrt{x}}=\dfrac{4\sqrt{x}}{\sqrt{x+3}}\Leftrightarrow x+3=4x\Leftrightarrow x=1\left(tm\right)\)
\(b.2x^4-5x^3+6x^2-5x+2=0\Leftrightarrow\left(x-1\right)^2\left(2x^2-2x+2\right)\Leftrightarrow\left[{}\begin{matrix}x=1\\2x^2-2x+2=0\left(vô-nghiệm\right)\end{matrix}\right.\)
\(a,ĐK:\left\{{}\begin{matrix}x\ge5\\x\le3\end{matrix}\right.\Leftrightarrow x\in\varnothing\)
Vậy pt vô nghiệm
\(b,ĐK:x\le\dfrac{2}{5}\\ PT\Leftrightarrow4-5x=2-5x\\ \Leftrightarrow0x=2\Leftrightarrow x\in\varnothing\)
\(c,ĐK:x\ge-\dfrac{3}{2}\\ PT\Leftrightarrow x^2+4x+5-2\sqrt{2x+3}=0\\ \Leftrightarrow\left(2x+3-2\sqrt{2x+3}+1\right)+\left(x^2+2x+1\right)=0\\ \Leftrightarrow\left(\sqrt{2x+3}-1\right)^2+\left(x+1\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}2x+3=1\\x+1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x=-1\\x=-1\end{matrix}\right.\Leftrightarrow x=-1\left(tm\right)\\ d,PT\Leftrightarrow\left|x-1\right|=\left|2x-1\right|\Leftrightarrow\left[{}\begin{matrix}x-1=2x-1\\x-1=1-2x\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{2}{3}\end{matrix}\right.\)
\(a,ĐK:x\ge-\dfrac{3}{2}\\ PT\Leftrightarrow2x+3=25\Leftrightarrow x=11\left(tm\right)\\ b,ĐK:x\ge2\\ PT\Leftrightarrow x^2+2x=2x+4\\ \Leftrightarrow x^2=4\Leftrightarrow\left[{}\begin{matrix}x=2\left(tm\right)\\x=-2\left(ktm\right)\end{matrix}\right.\Leftrightarrow x=2\)
a.\(\left(x^2+2x+5\right)\left(x^2+4x\right)=0\)
Ta có: \(x^2+2x+5=x^2+2x+1+4=\left(x+1\right)^2+4\ge4>0;\forall x\)
\(\Rightarrow x^2+4x=0\)
\(\Leftrightarrow x\left(x+4\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=-4\end{matrix}\right.\)
b.\(\left(x^2-4x+4\right)\left(x^2-3x\right)=0\)
\(\Leftrightarrow\left(x-2\right)^2x\left(x-3\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=0\\x=3\end{matrix}\right.\)
c.\(1,2x^3-x^2-0,2x=0\)
\(\Leftrightarrow x\left(1,2x^2-x-0,2\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=1\\x=-\dfrac{1}{6}\end{matrix}\right.\)
Theo em những bài toán khó lớp 9 hay lớp 10 mik mang lên H.VN nhé OLM ít ng trl lắm ạ