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\(\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1\) ( SỬA ĐỀ)
\(\sqrt{x-1-2.2.\sqrt{x-1}+4}+\sqrt{x-1-2.3.\sqrt{x-1}+9}=1\)
\(|x-1-2|+|x-1-3|=1\)
\(|x-3|+|x-4|=1\)
Với \(x\le3\)thì PT thành \(3-x+4-x=1\) \(\Rightarrow-2x=-6\Rightarrow x=3\)(thõa mãn)
Với \(3\le x< 4\)thì PT thành \(x-3+4-x=1\Leftrightarrow0x=0\Rightarrow\)Đúng với mọi x từ \(3\le x< 4\)
Với \(x\ge4\)thì PT thành \(x-3+x-4=1\Leftrightarrow2x=8\Leftrightarrow x=4\)(thõa mãn)
Vậy \(3\le x\le4\)
a/ ĐKXĐ: \(2\le x\le10\)
\(\Leftrightarrow\sqrt{x-2}+\sqrt{10-x}-x^2+12x-20-20=0\)
Đặt \(\sqrt{x-2}+\sqrt{10-x}=a>0\)
\(\Rightarrow a^2=8+2\sqrt{-x^2+12x-20}\Rightarrow-x^2+12x-20=\frac{\left(a^2-8\right)^2}{4}\)
Phương trình trở thành:
\(a+\frac{\left(a^2-8\right)^2}{4}-20=0\Leftrightarrow a^4-16a^2+4a-16=0\)
\(\Leftrightarrow a^2\left(a-4\right)\left(a+4\right)+4\left(a-4\right)=0\)
\(\Leftrightarrow\left(a-4\right)\left(a^3+4a^2+4\right)=0\)
\(\Leftrightarrow a=4\) (do \(a^3+4a^2+4>0\) \(\) \(\forall a>0\))
\(\Leftrightarrow\sqrt{x-2}+\sqrt{10-x}=4\)
Mà \(\sqrt{x-2}+\sqrt{10-x}\le\sqrt{2\left(x-2+10-x\right)}=4\)
Dấu "=" xảy ra khi và chỉ khi \(x-2=10-x\Leftrightarrow x=6\)
b/ ĐKXĐ:...
Ta có:
\(VT=1.\sqrt{x^2+x-1}+1.\sqrt{x-x^2+1}\le\frac{1+x^2+x-1}{2}+\frac{1+x-x^2+1}{2}=x+1\)
\(\Rightarrow x^2-x+2\le x+1\)
\(\Leftrightarrow x^2-2x+1\le0\)
\(\Leftrightarrow\left(x-1\right)^2\le0\Rightarrow x=1\)
Vậy pt có nghiệm duy nhất \(x=1\)
2,\(pt\Leftrightarrow12\left(\sqrt{x+1}-2\right)+x^2+x-12=0\)
\(\Leftrightarrow12\cdot\frac{x-3}{\sqrt{x+1}+2}+\left(x-3\right)\left(x+4\right)=0\)
\(\Leftrightarrow\left(x-3\right)\left(\frac{12}{\sqrt{x+1}+2}+x+4\right)=0\)
Vì \(\left(\frac{12}{\sqrt{x+1}+2}+x+4\right)\ge0\left(\forall x>-1\right)\)
\(\Rightarrow x=3\)
1/ Đặt \(\sqrt{x^2+2}=t>0\Rightarrow x^2=t^2-2\)
\(t^2-2+\left(3-t\right)x-1-2t=0\)
\(\Leftrightarrow t^2-2t-3-\left(t-3\right)x=0\)
\(\Leftrightarrow\left(t-3\right)\left(t+1\right)-\left(t-3\right)x=0\)
\(\Leftrightarrow\left(t-3\right)\left(t+1-x\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}t-3=0\\t+1-x=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}t=3\\t=x-1\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}\sqrt{x^2+2}=3\left(1\right)\\\sqrt{x^2+2}=x-1\left(2\right)\end{matrix}\right.\)
\(\left(1\right)\Leftrightarrow x^2=7\Rightarrow x=\pm\sqrt{7}\)
\(\left(2\right)\Leftrightarrow\left\{{}\begin{matrix}x-1\ge0\\x^2+2=\left(x-1\right)^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\x^2+2=x^2-2x+1\end{matrix}\right.\) \(\Rightarrow x=\dfrac{-1}{2}\left(l\right)\)
Vậy nghiệm pt là \(x=\pm\sqrt{7}\)
2/
\(x^2+3-6x\sqrt{x^2+3}+9x^2-\sqrt{x^2+3}+3x-2=0\)
\(\Leftrightarrow\left(\sqrt{x^2+3}-3x\right)^2-\left(\sqrt{x^2+3}-3x\right)-2=0\)
Đặt \(\sqrt{x^2+3}-3x=t\)
\(\Rightarrow t^2-t-2=0\) \(\Rightarrow\left[{}\begin{matrix}t=-1\\t=2\end{matrix}\right.\)
TH1: \(\sqrt{x^2+3}-3x=-1\Rightarrow\sqrt{x^2+3}=3x-1\)
\(\Leftrightarrow\left\{{}\begin{matrix}3x-1\ge0\\x^2+3=\left(3x-1\right)^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{1}{3}\\8x^2-6x-2=0\end{matrix}\right.\) \(\Rightarrow x=1\)
TH2: \(\sqrt{x^2+3}-3x=2\Leftrightarrow\sqrt{x^2+3}=3x+2\)
\(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{-2}{3}\\x^2+3=\left(3x+2\right)^2\end{matrix}\right.\) \(\Leftrightarrow\left\{{}\begin{matrix}x\ge\dfrac{-2}{3}\\8x^2+12x+1=0\end{matrix}\right.\) \(\Rightarrow x=\dfrac{-3+\sqrt{7}}{4}\)
3/ ĐKXĐ: \(\dfrac{3}{2}\le x\le\dfrac{5}{2}\)
\(1.\sqrt{2x-3}+1.\sqrt{5-2x}\le\sqrt{\left(1^2+1^2\right)\left(2x-3+5-2x\right)}=2\)
\(\Rightarrow VT\le2\)
\(VP=3\left(x^2-4x+4\right)+2=3\left(x-2\right)^2+2\ge2\)
\(\Rightarrow VT=VP\Leftrightarrow\left\{{}\begin{matrix}x-2=0\\2x-3=5-2x\end{matrix}\right.\) \(\Rightarrow x=2\)
Vậy pt có nghiệm duy nhất \(x=2\)
4/
ĐKXĐ: \(x\ge\dfrac{-5}{4}\)
\(x^2-2x+1+4x+5-6\sqrt{4x+5}+9=0\)
\(\Leftrightarrow\left(x-1\right)^2+\left(\sqrt{4x+5}-3\right)^2=0\)
\(\Leftrightarrow\left\{{}\begin{matrix}x-1=0\\\sqrt{4x+5}-3=0\end{matrix}\right.\) \(\Rightarrow x=1\)
Vậy pt có nghiệm duy nhất \(x=1\)
a) điều kiện : \(x\ge3\)
ta có : \(\sqrt{x^2-9}-\sqrt{4x-12}\le0\) \(\Leftrightarrow\sqrt{x^2-9}\le\sqrt{4x-12}\)
\(\Leftrightarrow x^2-9\le4x-12\Leftrightarrow x^2-4x+3\le0\Leftrightarrow\left(x-1\right)\left(x-3\right)\le0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x-1\ge0\\x-3\le0\end{matrix}\right.\\\left\{{}\begin{matrix}x-1\le0\\x-3\ge0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}1\le x\le3\\x\in\varnothing\end{matrix}\right.\) kết hợp với điều kiện \(\Rightarrow x=3\)
b) điều kiện \(x\ge1\)
ta có : \(\sqrt{x^2-1}-\sqrt{x-1}>0\) \(\Leftrightarrow\sqrt{x^2-1}>\sqrt{x-1}\)
\(\Leftrightarrow x^2-1>x-1\Leftrightarrow x^2-x< 0\Leftrightarrow x\left(x-1\right)< 0\)
\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}x>0\\x-1>0\end{matrix}\right.\\\left\{{}\begin{matrix}x< 0\\x-1< 0\end{matrix}\right.\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x>1\\x< 0\left(L\right)\end{matrix}\right.\) vậy \(x>1\)
c) điều kiện \(x\ge3\)
ta có : \(\sqrt{2x^2-12x+18}+\sqrt{x-3}>0\)
\(\Leftrightarrow\sqrt{2\left(x-3\right)^2}+\sqrt{x-3}>0\) \(\Rightarrow x\ne3\) kết hợp với điều kiện \(\Rightarrow x>3\)