Lời giải:
PT $\Rightarrow x^2+1-x\sqrt{x^2+1}=5$
$\Leftrightarrow 2x^2-2x\sqrt{x^2+1}-8=0$
$\Leftrightarrow x^2+(x^2+1)-2x\sqrt{x^2+1}-9=0$
$\Leftrightarrow (x-\sqrt{x^2+1})^2-9=0$
$\Leftrightarrow (x-\sqrt{x^2+1}-3)(x-\sqrt{x^2+1}+3)=0$
Xét các TH:
$x-3=\sqrt{x^2+1}$
$\Rightarrow x\geq 3$ và $(x-3)^2=x^2+1$
$\Rightarrow x=\frac{4}{3}$ (vô lý)
$x+3=\sqrt{x^2+1}$
$\Rightarrow x\geq -3$ và $(x+3)^2=x^2+1$
$\Rightarrow x=\frac{-4}{3}$
Vậy...........
giải pt
a) \(3\sqrt{x}+\frac{3}{2\sqrt{x}}=2x+\frac{1}{2x}-7\)
b) \(5\sqrt{x}+\frac{5}{2\sqrt{x}}=2x+\frac{1}{2x}+4\)
c) \(\sqrt{2x^2+8x+5}+\sqrt{2x^2-4x+5}=6\sqrt{x}\)
d) \(x+1+\sqrt{x^2-4x+1}=3\sqrt{x}\)
e) \(x^2+2x\sqrt{x-\frac{1}{x}}=3x+1\)
f) \(x^2-6x+x\sqrt{\frac{x^2-6}{x}}-6=0\)
g) \(\frac{3x^2}{3+\sqrt{x}}+6+2\sqrt{x}=5x\)
h) \(\frac{x^2}{4-3\sqrt{x}}+8=3\left(x+2\sqrt{x}\right)\)
giải pt
a) \(2\sqrt{x+2+2\sqrt{x+1}}-\sqrt{x+1}=4\)
b) \(\sqrt{x-2\sqrt{x-1}}+\sqrt{x+3-4\sqrt{x-1}}=1\)
c) \(\sqrt{x+2\sqrt{x-1}}-\sqrt{x-2\sqrt{x-1}}=2\)
d) \(\sqrt{\frac{5}{4}-x^2+\sqrt{1-x^2}}+\sqrt{\frac{5}{4}-x^2-\sqrt{1-x^2}}=x+1\)
1)\(\begin{cases}x+\sqrt{x^2+1}=y+\sqrt{y^2-1}\left(1\right)\\3\sqrt{y-1}+\sqrt{x}=2\sqrt{y+1}\left(2\right)\end{cases}\) nhân liên hợp pt 1 đc (\(\left(x^2-y^2+1\right)\left(\frac{1}{x+\sqrt{y^2-1}}+\frac{1}{\sqrt{x^2+1}+y}\right)\) thì TH1 \(x^2-y^2+1\) lm ntn
2\(\begin{cases}\sqrt{x^2+xy+2y^2}+\sqrt{xy}=3y\\\sqrt{y-1}+\sqrt{x-1}+x+y=6\end{cases}\)
3\(\begin{cases}\frac{\sqrt{x^2+5}}{x}+\frac{\sqrt{y^2+3}}{y}=\frac{7}{2}\\x\sqrt{x^2+5}+y\sqrt{y^2+3}=3+x^2+y^2\end{cases}\)
Giải pt sau :
1, \(\sqrt{x+1}+\sqrt{4-x}+\sqrt{\left(x+1\right)\left(4-x\right)}=5\)
2, \(\sqrt{x+4}+\sqrt{x-4}=2x-12+2\sqrt{x^2-16}\)
3, \(\sqrt{x+\sqrt{6x-9}}+\sqrt{x-\sqrt{6x-9}}=\sqrt{6}\)
4, \(\frac{4}{x+\sqrt{x^2+x}}-\frac{1}{x-\sqrt{x^2+x}}=\frac{3}{x}\)
5, \(\sqrt{x^2+x+4}+\sqrt{x^2+x+1}=\sqrt{2x^2+2x+9}\)
giải pt
a) \(2\sqrt{\frac{x}{x-1}}-\sqrt{\frac{x-1}{x}}=\frac{5x-2}{x}\)
b) \(3\sqrt{\frac{2x}{x-1}}+4\sqrt{\frac{x-1}{2x}}=\frac{5x-3}{2x}+9\)
c) \(\sqrt{\frac{x}{3-2x}}+5\sqrt{\frac{3-2x}{x}}=\frac{12-9x}{x}+6\)
d) \(\frac{x-1}{x}-2\sqrt{\frac{x-1}{x}}=3\)
e) \(\sqrt{\frac{x}{x-1}}+\sqrt{\frac{x-1}{x}}=\frac{3}{\sqrt{2}}\)
f) \(\sqrt{x-\frac{1}{x}}=\frac{1}{\sqrt{x}}-\sqrt{x}\)
giai pt:
a) \(\frac{3x+\sqrt{x^2-x-1}}{x+1}=\frac{7}{3}\)
b) \(\frac{2}{2\sqrt{x^2-2x+1}}=\frac{1}{x-1}\)
c) \(\frac{6}{6-\sqrt{x}}+\frac{1}{\sqrt{x}}=1\)
d) \(\frac{2}{\sqrt{x-1}}+\sqrt{x-1}=\frac{3\sqrt{x-1}+1}{\sqrt{x-1}}-1\)
e) \(\sqrt{x+3-\sqrt{x-1}=2}\)
f) \(\sqrt{x^3+x^2+6x+28}=x+5\)
g) \(\sqrt{x^4-4x^3+14x-11}=1-x\)
Giải hệ phương trình: \(\left\{{}\begin{matrix}\frac{1}{\sqrt{4x^2+8x+5}}+\frac{1}{\sqrt{4y^2-8x+5}}=\frac{2}{\sqrt{\left(x+y\right)^2+1}}\\\frac{1}{\sqrt{x-1}}+\frac{1}{\sqrt{y-3}}=\frac{2\sqrt{5}}{5}\end{matrix}\right.\)
1. \(x^3-x^2+12x\sqrt{x-1}+20=0\)
2. \(x^3+\sqrt{\left(x-1\right)^3}=9x+8\)
3. \(\sqrt{2x^2+x+1}+\sqrt{x^2-x+1}=3x\)
4. \(x^6+\left(x^3-3\right)^3=3x^5-9x^2-1\)
5. \(x^2-6\left(x+3\right)\sqrt{x+1}+14x+3\sqrt{x+1}+13=0\)
6. \(x^2-4x+\left(x-3\right)\sqrt{x^2-x+1}=-1\)
7. \(\sqrt{2x-1}+\sqrt{5-x}=x-2+2\sqrt{-2x^2+11x-5}\)
8. \(\sqrt{5x+11}-\sqrt{6-x}+5x^2-14x-60=0\)
9. \(x^2+6x+8=3\sqrt{x+2}\)
10. \(2x^2+3x-2=\left(2x-1\right)\sqrt{2x^2+x-3}\)
11. \(\sqrt{x+1}+\sqrt{4-x}-\sqrt{\left(x+1\right)\left(4-x\right)}=1\)
12. \(x^2-\sqrt{x^2-4x}=4\left(x+3\right)\)
13. \(x^2-x-4=2\sqrt{x-1}\left(1-x\right)\)
14. \(\frac{1}{\sqrt{x}+1}+\frac{1}{\sqrt{x}-1}=1\)
15. \(\sqrt{2x^2+3x+2}+\sqrt{4x^2+6x+21}=11\)
16. \(\sqrt{x+3+3\sqrt{2x-3}}+\sqrt{x-1+\sqrt{2x-1}}=2\sqrt{2}\)
17. \(\left(x-2\right)^2\left(x-1\right)\left(x-3\right)=12\)
18. \(2x^2+\sqrt{x^2-2x-19}=4x+74\)
19. \(x^4+x^2-20=0\)
20. \(x+\sqrt{4-x^2}=2+3x\sqrt{4-x^2}\)
21. \(\left(x^2+x+1\right)\left(\sqrt[3]{\left(3x-2\right)^2}+\sqrt[3]{3x-2}+1\right)=9\)
22. \(\sqrt{x^2-3x+5}+x^2=3x+7\)
23. \(x^2+6x+5=\sqrt{x+7}\)
24. \(\frac{2x^2-3x+10}{x+2}=3\sqrt{\frac{x^2-2x+4}{x+2}}\)
25. \(5\sqrt{x-1}-\sqrt{x+7}=3x-4\)
26. \(2\left(x^2+2\right)=5\sqrt{x^3+1}\)
27. \(\sqrt{x-1}+\sqrt{5-x}-2=2\sqrt{\left(x-1\right)\left(5-x\right)}\)
28. \(x^2+\frac{9x^2}{\left(x-3\right)^2}=40\)
29. \(\frac{26x+5}{\sqrt{x^2+30}}+2\sqrt{26x+5}=3\sqrt{x^2+30}\)
30. \(\frac{\sqrt{27+x^2+x}}{2+\sqrt{5-\left(x^2+x\right)}}=\frac{\sqrt{27+2x}}{2+\sqrt{5-2x}}\)
giải pt
a) \(\sqrt{2x^2+5x+2}-2\sqrt{2x^2+5x-6}=0\)
b) \(\sqrt[5]{\frac{16x}{x-1}}+\sqrt[5]{\frac{x-1}{16x}}=\frac{5}{2}\)
c) \(\sqrt{6x^2-12x+7}+2x=x^2\)
d) \(x\left(x+1\right)-\sqrt{x^2+x+4}+2=0\)
e) \(\sqrt{3x^2+6x+4}=2-2x-x^2\)
Giải các phương trình sau:
a)\(\frac{2x-5}{2x^2+3x-5}+\frac{3x+1}{1-x}=\frac{x+20}{4x+10}\)
b)\(\sqrt{5x+1}=\sqrt{14x+7}+\sqrt{2x+3}\)
c)\(\sqrt{x+3}-\sqrt{x+1}=\sqrt{5x+7}\)
d)\(\sqrt{\left(x-3\right)^2\left(x-1\right)}=x-3\)
e)\(\sqrt{x^2+2x+2}=1-x\)
f)\(\sqrt{x^4+x^2+4}=x^2+2\)
g)\(2x^2-6x+1=\sqrt{4x+5}\)
h)\(x^2+\sqrt{x+11}=11\)
