`(2\sqrt{x}-2)/(\sqrt{x}-3) <0` (ĐKXĐ: `x>=0`)

Dễ thấy `2\sqrt{x}-2 > \sqrt{x}-3`

Để `(2\sqrt{x}-2)/(\sqrt{x}-3) <0`

`⇔ {(2\sqrt{x}-2>0),(\sqrt{x}-3<0):} ...

`{(x-2y=5),(3x-6y=15):}`

`⇔ {(x-2y=5),(x-2y=5):}`

`⇒` hệ có vô số nghiệm

Vậy,..

`{(-3x+y=4),(6x-2y=7):}`

`= {(-3x+y=4),(-3x+y=-7/2):}` (v&o ...

`x^3+y^3-z^3+3xyz`

`= (x+y)^3 - z^3 - 3xy(x+y) + 3xyz`

`= (x+y-z)(x^2+y^2+z^2+2xy+xyz) - 3xy(x+y-z)`

`= (x+y-z)(x^2+y^2+z^2-xy+xyz)`

`color{purple}{@SHIN}`

`a, (4x+8)(x-2)`

`= 4x^2 - 8x + 8x - 16 = 4x^2-16`

`b, (3x-3)(2x+1)`

`= 6x^2 + 3x - 6x - 3`

`= 6x^2 - 3x - 3`

`c, (x+2)(x-3)`

`= x^2 - 3x + 2x - 6`

...

Vì `n+3 \vdots n^2 - 7`

`⇒ (n+3)(n-3) \vdots n^2-7`

`⇒ n^2 - 9 \vdots n^2 - 7`

`⇒ -2 \vdots n^2 -7`

`⇒ n^2-7 ∈ Ư(-2)={-1;-2;1;2}`

...

`29/15 . 3/7 + 29/15 : 7/4`

`= 29/15 . 3/7 + 29/15 . 4/7`

`= 29/15 . (3/7 + 4/7)`

`= 29/15 . 1 = 29/15`

`color{purple}{@SHIN}`

`x^3 + 3x +2 =(x+2)\sqrt{x^3+2x+1}` (ĐkXĐ: `x>=-1`)

`⇔ x^3 + 2x + 1 - (x+2)\sqrt{x^3+2x+1} + (x + 2) - 1 =0`

Đặt `a=\sqrt{x^3+2x+1}; b=x+2;`          ...

`a, A=3-x+\sqrt{x^2+4x+4}`

`A=3-x+\sqrt{(x+2)^2}`

+, Nếu `x+2 >= 0`

`⇒ A= 3 - x + x + 2`

`A=5  ∀ x`

+, Nếu `x+2<0`

`⇒ A=3-x-x-2 ...

`a, (x-2)\sqrt{2x^2-x-10}=4-x^2`          (ĐKXĐ: `x <= 2, x >= 5/2`)

`⇔ (x-2)\sqrt{2x^2-x-10}+(x-2)(x+2)=0`

`⇔ (x-2)(\sqrt{2x^2-x-10}+x+2)=0`< ...

`A=(a+b)/(\sqrt{a(3a+b}+\sqrt{b(3b+a)})`

`A=(4a+4b)/(2\sqrt{4a(3a+b}+2\sqrt{4b(3b+a)})`

Áp dụng BDT AM-GM

`2\sqrt{4a(3a+b)} <= 4a + (3a+b)`

`2\sqrt{4b(3b+a)} & ...