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NV
Nguyễn Việt Lâm
Giáo viên
11 tháng 1 2019
1/ \(\dfrac{5}{3}\le x\le\dfrac{7}{3}\)
Đặt \(\left\{{}\begin{matrix}\sqrt{3x-5}=a>0\\\sqrt{7-3x}=b>0\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a^2+b^2=2\\17-6x=2b^2+3\\6x-7=2a^2+3\end{matrix}\right.\)
Mặt khác theo BĐT Bunhiacốpxki:
\(a+b=\sqrt{3x-5}+\sqrt{7-3x}\le\sqrt{\left(1+1\right)\left(3x-5+7-3x\right)}=2\)
\(\Rightarrow0< a+b\le2\)
Ta được hệ pt:
\(\left\{{}\begin{matrix}a^2+b^2=2\\\left(2b^2+3\right).a+\left(2a^2+3\right)b=2+8ab\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}\left(a+b\right)^2-2ab=2\\2ab^2+3a+2a^2b+3b-8ab-2=0\end{matrix}\right.\)
\(\Leftrightarrow\left\{{}\begin{matrix}2ab=\left(a+b\right)^2-2\\2ab\left(a+b\right)+3\left(a+b\right)-8ab-2=0\end{matrix}\right.\)
\(\Rightarrow\left(\left(a+b\right)^2-2\right)\left(a+b\right)+3\left(a+b\right)-4\left(a+b\right)^2+6=0\)
\(\Leftrightarrow\left(a+b\right)^3-4\left(a+b\right)^2+\left(a+b\right)+6=0\)
\(\Rightarrow\left[{}\begin{matrix}a+b=-1< 0\left(l\right)\\a+b=2\\a+b=3>2\left(l\right)\end{matrix}\right.\)
\(\Rightarrow a+b=2\) , dấu "=" xảy ra khi và chỉ khi:
\(3x-5=7-3x\Rightarrow x=2\)
Vậy pt có nghiệm duy nhất \(x=2\)
NV
Nguyễn Việt Lâm
Giáo viên
11 tháng 1 2019
2/ ĐKXĐ: \(x\ne\pm2\)
\(\left(\dfrac{x-1}{x+2}\right)^2+4\left(\dfrac{x+1}{x-2}\right)^2-\left(\dfrac{15}{x^2-4}+5\right)=0\)
\(\Leftrightarrow\left(\dfrac{x-1}{x+2}\right)^2+4\left(\dfrac{x+1}{x-2}\right)^2-5.\left(\dfrac{x^2-1}{x^2-4}\right)=0\)
\(\Leftrightarrow\left(\dfrac{x-1}{x+2}\right)^2-\left(\dfrac{x^2-1}{x^2-4}\right)-4\left[\left(\dfrac{x^2-1}{x^2-4}\right)-\left(\dfrac{x+1}{x-2}\right)^2\right]=0\)
\(\Leftrightarrow\left(\dfrac{x-1}{x+2}\right)\left(\dfrac{x-1}{x+2}-\dfrac{x+1}{x-2}\right)-4\left(\dfrac{x+1}{x-2}\right)\left(\dfrac{x-1}{x+2}-\dfrac{x+1}{x-2}\right)=0\)
\(\Leftrightarrow\left(\dfrac{x-1}{x+2}-\dfrac{4\left(x+1\right)}{x-2}\right)\left(\dfrac{x-1}{x+2}-\dfrac{x+1}{x-2}\right)=0\)
\(\Leftrightarrow\left[{}\begin{matrix}\dfrac{x-1}{x+2}=\dfrac{4\left(x+1\right)}{x-2}\\\dfrac{x-1}{x+2}=\dfrac{x+1}{x-2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x^2-3x+2=4\left(x^2+3x+2\right)\\x^2-3x+2=x^2+3x+2\end{matrix}\right.\)
\(\Leftrightarrow\left[{}\begin{matrix}3x^2+15x+6=0\\6x=0\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=0\\x=\dfrac{-5+\sqrt{17}}{2}\\x=\dfrac{-5-\sqrt{17}}{2}\end{matrix}\right.\)
AH
Akai Haruma
Giáo viên
14 tháng 10 2018
1)
ĐK: \(x\geq 5\)
PT \(\Leftrightarrow \sqrt{4(x-5)}+3\sqrt{\frac{x-5}{9}}-\frac{1}{3}\sqrt{9(x-5)}=6\)
\(\Leftrightarrow \sqrt{4}.\sqrt{x-5}+3\sqrt{\frac{1}{9}}.\sqrt{x-5}-\frac{1}{3}.\sqrt{9}.\sqrt{x-5}=6\)
\(\Leftrightarrow 2\sqrt{x-5}+\sqrt{x-5}-\sqrt{x-5}=6\)
\(\Leftrightarrow 2\sqrt{x-5}=6\Rightarrow \sqrt{x-5}=3\Rightarrow x=3^2+5=14\)
AH
Akai Haruma
Giáo viên
14 tháng 10 2018
2)
ĐK: \(x\geq -1\)
\(\sqrt{x+1}+\sqrt{x+6}=5\)
\(\Leftrightarrow (\sqrt{x+1}-2)+(\sqrt{x+6}-3)=0\)
\(\Leftrightarrow \frac{x+1-2^2}{\sqrt{x+1}+2}+\frac{x+6-3^2}{\sqrt{x+6}+3}=0\)
\(\Leftrightarrow \frac{x-3}{\sqrt{x+1}+2}+\frac{x-3}{\sqrt{x+6}+3}=0\)
\(\Leftrightarrow (x-3)\left(\frac{1}{\sqrt{x+1}+2}+\frac{1}{\sqrt{x+6}+3}\right)=0\)
Vì \(\frac{1}{\sqrt{x+1}+2}+\frac{1}{\sqrt{x+6}+3}>0, \forall x\geq -1\) nên $x-3=0$
\(\Rightarrow x=3\) (thỏa mãn)
Vậy .............
PA
16 tháng 8 2017
\(\sqrt{x^2-3x+2}-\sqrt{x+3}=\sqrt{x-2}+\sqrt{x^2+2x-3}\)
\(\Leftrightarrow\left(\sqrt{x^2-3x+2}-\sqrt{x-2}\right)-\left(\sqrt{x^2+2x-3}+\sqrt{x+3}\right)=0\)
\(\Leftrightarrow\dfrac{\left(x^2-3x+2\right)-\left(x-2\right)}{\sqrt{x^2-3x+2}+\sqrt{x-2}}-\dfrac{\left(x^2+2x-3\right)-\left(x+3\right)}{\sqrt{x^2+2x-3}-\sqrt{x+3}}=0\)
\(\Leftrightarrow\dfrac{\left(x-2\right)^2}{\sqrt{\left(x-2\right)\left(x-1\right)}+\sqrt{x-2}}-\dfrac{\left(x-2\right)\left(x+3\right)}{\sqrt{\left(x+3\right)\left(x-1\right)}-\sqrt{x+3}}=0\)
\(\Leftrightarrow\left(x-2\right)\left[\dfrac{x-2}{\sqrt{x-2}\left(\sqrt{x-1}+1\right)}-\dfrac{x+3}{\sqrt{x+3}\left(\sqrt{x-1}-1\right)}\right]=0\)
\(\Leftrightarrow\left(x-2\right)\left[\dfrac{\sqrt{x-2}}{\sqrt{x-1}+1}-\dfrac{\sqrt{x+3}}{\sqrt{x-1}-1}\right]=0\)
Pt \(\dfrac{\sqrt{x-2}}{\sqrt{x-1}+1}-\dfrac{\sqrt{x+3}}{\sqrt{x-1}-1}=0\) vô no
(vì \(\dfrac{\sqrt{x-2}}{\sqrt{x-1}+1}< \dfrac{\sqrt{x+3}}{\sqrt{x-1}-1}\forall x\ge2\Rightarrow VT< 0\))
=> x - 2 = 0
<=> x = 2 (nhận)
PA
16 tháng 8 2017
\(\sqrt{4x+1}-\sqrt{3x-2}=\dfrac{x+3}{5}\)
\(\Leftrightarrow\dfrac{\left(4x+1\right)-\left(3x-2\right)}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{x+3}{5}=0\)
\(\Leftrightarrow\dfrac{x+3}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{x+3}{5}=0\)
\(\Leftrightarrow\left(\dfrac{1}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{1}{5}\right)\left(x+3\right)=0\)
TH1:
x + 3 = 0
<=> x = - 3 (loại)
TH2:
\(\dfrac{1}{\sqrt{4x+1}+\sqrt{3x-2}}-\dfrac{1}{5}=0\)
\(\Leftrightarrow\sqrt{4x+1}+\sqrt{3x-2}=5\)
\(\Leftrightarrow\left(\sqrt{4x+1}-3\right)+\left(\sqrt{3x-2}-2\right)=0\)
\(\Leftrightarrow\dfrac{4x+1-9}{\sqrt{4x+1}+3}+\dfrac{3x-2-4}{\sqrt{3x-2}+2}=0\)
\(\Leftrightarrow\dfrac{4\left(x-2\right)}{\sqrt{4x+1}+3}+\dfrac{3\left(x-2\right)}{\sqrt{3x-2}+2}=0\)
\(\Leftrightarrow\left(\dfrac{4}{\sqrt{4x+1}+3}+\dfrac{3}{\sqrt{3x-2}+2}\right)\left(x-2\right)=0\)
Pt \(\dfrac{4}{\sqrt{4x+1}+3}+\dfrac{3}{\sqrt{3x-2}+2}>0\forall x\ge\dfrac{2}{3}\) => vô no
=> x - 2 = 0
<=> x = 2 (nhận)
~ ~ ~
Vậy x = 2
\(\dfrac{3x}{x^2-4x+7}+\dfrac{2x}{x^2-6x+7}=2\) (x \(\ne\) 3 + \(\sqrt{2}\); x \(\ne\) 3 - \(\sqrt{2}\))
Đặt x2 - 5x + 7 = t (t \(\ne\) \(\pm\) x)
Khi đó:
\(\dfrac{3x}{t+x}+\dfrac{2x}{t-x}=2\)
\(\Leftrightarrow\) \(\dfrac{3x\left(t-x\right)+2x\left(t+x\right)}{t^2-x^2}=2\)
\(\Leftrightarrow\) 3xt - 3x2 + 2xt + 2x2 = 2(t2 - x2)
\(\Leftrightarrow\) 5xt - x2 = 2t2 - 2x2
\(\Leftrightarrow\) 2t2 - x2 - 5xt = 0
\(\Leftrightarrow\) 2(t2 - \(\dfrac{5}{2}\)xt + \(\dfrac{25}{16}\)x2 - \(\dfrac{33}{16}\)x2) = 0
\(\Leftrightarrow\) (t - \(\dfrac{5}{4}\))2 - \(\dfrac{33}{16}\)x2 = 0
\(\Leftrightarrow\) (t - \(\dfrac{5}{4}\) - \(\dfrac{\sqrt{33}}{4}\))(t - \(\dfrac{5}{4}\) + \(\dfrac{\sqrt{33}}{4}\)) = 0
\(\Leftrightarrow\) \(\left[{}\begin{matrix}t=\dfrac{5+\sqrt{33}}{4}\\t=\dfrac{5-\sqrt{33}}{4}\end{matrix}\right.\)
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x^2-5x+7=\dfrac{5+\sqrt{33}}{4}\\x^2-5x+7=\dfrac{5-\sqrt{33}}{4}\end{matrix}\right.\)
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x^2-2.\dfrac{5}{2}x+\dfrac{25}{4}+\dfrac{3}{4}=\dfrac{5+\sqrt{33}}{4}\\x^2-2.\dfrac{5}{2}+\dfrac{25}{4}+\dfrac{3}{4}=\dfrac{5-\sqrt{33}}{4}\end{matrix}\right.\)
\(\Leftrightarrow\) \(\left[{}\begin{matrix}\left(x-\dfrac{5}{2}\right)^2=\dfrac{2+\sqrt{33}}{4}\\\left(x-\dfrac{5}{2}\right)^2=\dfrac{2-\sqrt{33}}{4}\end{matrix}\right.\)
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x-\dfrac{5}{2}=\dfrac{\sqrt{2+\sqrt{33}}}{2}\\x-\dfrac{5}{2}=\dfrac{\sqrt{2-\sqrt{33}}}{2}\end{matrix}\right.\)
\(\Leftrightarrow\) \(\left[{}\begin{matrix}x=\dfrac{\sqrt{2+\sqrt{33}}+5}{2}\left(TM\right)\\x=\dfrac{\sqrt{2-\sqrt{33}}+5}{2}\left(KTM\right)\end{matrix}\right.\)
Vậy S = {\(\dfrac{\sqrt{2+\sqrt{33}}+5}{2}\)}
Chúc bn học tốt! (Ko bt đúng ko nhưng nhìn số ko đẹp lắm :v)
ĐKXĐ: ....
Nhận thấy \(x=0\) không phải nghiệm, pt tương đương:
\(\dfrac{3}{x+\dfrac{7}{x}-4}+\dfrac{2}{x+\dfrac{7}{x}-6}=2\)
Đặt \(x+\dfrac{7}{x}-6=t\)
\(\Rightarrow\dfrac{3}{t+2}+\dfrac{2}{t}=2\Leftrightarrow3t+2\left(t+2\right)=2t\left(t+2\right)\)
\(\Leftrightarrow2t^2-t-4=0\)
\(\Leftrightarrow...\)