ĐK:\(x\ge0\)
Đặt \(t=\sqrt{x+2}\left(t\ge\sqrt{2}\right)\):
\(\frac{1}{\sqrt{t^2+2}+t}+\frac{1}{\sqrt{t^2-2}+t}=1\)
\(\frac{\sqrt{t^2+2}-t}{2}+\frac{t-\sqrt{t^2-2}}{2}=1\)
\(\Leftrightarrow\sqrt{t^2+2}-\sqrt{t^2-2}=2\)
Đến đây thì bình phương thôi.
giải pt
a) \(\sqrt{2x+3}+\sqrt{4-x}=6x-3\left(\sqrt{2x+3}-\sqrt{4-x}\right)^2-10\)
b) \(\sqrt{4x+1}+2\sqrt{1-x}+10\sqrt{-4x^2+3x+1}=13\)
c) \(\left(x^2+1\right)^2=13-x\sqrt{2x^2+4}\)
d) \(\left(\sqrt{x+1}+\sqrt{x-1}\right)^2-3=\frac{1}{\sqrt{x+1}-\sqrt{x-1}}\)
e) \(\left(\frac{2x-3}{\sqrt{x^2-1}}+2\right)\left(\frac{1}{\sqrt{x-1}}-\frac{1}{\sqrt{x+1}}\right)=\frac{1}{x^2-1}\)
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) \(\sqrt{x-\frac{1}{x}}=\frac{1}{\sqrt{x}}-\sqrt{x}\)
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) \(2\sqrt{x+2+2\sqrt{x+1}}-\sqrt{x+1}=4\)
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) \(\sqrt{\frac{5}{4}-x^2+\sqrt{1-x^2}}+\sqrt{\frac{5}{4}-x^2-\sqrt{1-x^2}}=x+1\)
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) \(\frac{x^2}{4-3\sqrt{x}}+8=3\left(x+2\sqrt{x}\right)\)
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 pt
a) \(\sqrt{x+3}=3-\sqrt{6-x}\)
b) \(\sqrt{3x-2}-\sqrt{x-7}=1\)
c) \(\frac{1-\sqrt{3x+1}}{\sqrt{x-1}-7}=1\)
d) \(\frac{x}{\sqrt{7x-4}-3}=\frac{x}{\sqrt{x+1}}\)
e) \(\sqrt{3x-2}-\sqrt{x-7}=1\)
f) \(2\sqrt{\frac{3x+1}{2x-1}}-\sqrt{\frac{x-1}{2x-1}}=2\)
\(8\sqrt{x^2+\frac{3}{4}+\sqrt{4x^2-1}}=10\sqrt{x^2+\frac{3}{4}-\sqrt{x^2}+\frac{1}{2}}+7\)
giải các hệ phương trình sau :
\(\frac{\sqrt{2\left(x^2-16\right)}}{\sqrt{x-3}}+\sqrt{x-3}=\frac{7-x}{\sqrt{x-3}}\)
\(\sqrt{x}+\sqrt{\frac{x^3+1}{x}}=\sqrt{x+1}+\sqrt{x^2-x+1}\)
\(\sqrt{\frac{x^3+1}{x+3}}+\sqrt{x+1}=\sqrt{x^2-x+1}+\sqrt{x+3}\)
