\(=\dfrac{y}{x^2}\cdot\dfrac{x}{y^2}=\dfrac{1}{xy}\)
\(=\dfrac{y}{x^2}\cdot\dfrac{x}{y^2}=\dfrac{1}{xy}\)
Cho \(x,y,z\ne0\) .Chứng minh rằng \(\dfrac{x^2}{y^2}+\dfrac{y^2}{z^2}+\dfrac{z^2}{x^2}\ge\dfrac{x}{z}+\dfrac{y}{x}+\dfrac{z}{y}\)
1/Tìm x
\(\dfrac{3}{x-3}-\dfrac{6x}{9-x^2}+\dfrac{x}{x+3}=0\)
2/ Cho \(\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}=1\)
Tính S = \(\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}\)
Chứng minh:
\(\dfrac{y-z}{\left(x-y\right)\left(x-z\right)}+\dfrac{z-x}{\left(y-z\right)\left(y-x\right)}+\dfrac{x-y}{\left(z-x\right)\left(z-y\right)}=\dfrac{2}{x-y}+\dfrac{2}{y-z}+\dfrac{2}{z-x}\)
Bài 4: Chứng minh
\(\dfrac{y-z}{\left(x-y\right)\left(x-z\right)}+\dfrac{z-x}{\left(y-z\right)\left(y-x\right)}+\dfrac{x-y}{\left(z-x\right)\left(z-y\right)}=\dfrac{2}{x-y}+\dfrac{2}{y-z}+\dfrac{2}{z-x}\)
Tinh:
\(\dfrac{2xy}{x^2-y^2}+\dfrac{x-y}{2x+2y}+\dfrac{y}{y-x}\)
Cho \(P=\dfrac{x}{y+z}+\dfrac{y}{z+x}+\dfrac{z}{x+y}\)
và \(Q=\dfrac{x^2}{y+z}+\dfrac{y^2}{z+x}+\dfrac{z^2}{x+y}\)
Chứng minh nếu P=1 thì Q=0
Cho x, y, z khác 0 thỏa mãn:
x(\(x^2-\dfrac{1}{y}-\dfrac{1}{z}\)) + y(\(y^2-\dfrac{1}{z}-\dfrac{1}{x}\)) + z(\(z^2-\dfrac{1}{x}-\dfrac{1}{y}\)) = 3
Tính : \(\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}\)
Thực hiện phép tính:
1. \(\dfrac{x^2}{x+1}+\dfrac{2x}{x^2-1}-\dfrac{1}{1-x}+1\)
2. \(\dfrac{1}{x^3-x}-\dfrac{1}{\left(x-1\right)x}+\dfrac{2}{x^2-1}\)
3. \(\dfrac{y}{xy-5y^2}-\dfrac{15y-25x}{y^2-25x^2}\)
4. \(\dfrac{4-2x+x^2}{2+x}-2-x\)
5. \(\dfrac{2x^3-2y^3}{3x+3y}:\dfrac{2x^2+2xy+y^2}{x^2+2xy+y^2}\)
6. \(\left(\dfrac{1+x}{1-x}-\dfrac{1-x}{1+x}\right)\left(\dfrac{3}{4x}+\dfrac{x}{4}-x\right)\)
( Tìm x,y,z biết : [ a,b,c,d] )
a) \(\dfrac{x}{2}=\dfrac{y}{3}\) và xy = 54
b) \(\dfrac{x}{5}=\dfrac{y}{3}\), \(x^2-y^2=4\) với x,y > 0
c) \(\dfrac{x}{2}=\dfrac{y}{3};\dfrac{y}{5}=\dfrac{z}{7}\) và \(x+y+z=92\)