Đáp án:

1. 

a, `(x + 1)^2 - 1 = 0`

`<=> (x + 1)^2 = 1`

<=>  \(\left[ \begin{array}{l}x + 1 = 1\\x + 1 = -1\end{array} \right.\) 

<=> \(\left[ \begin{array}{l}x=0\\x=-2\end{array} \right.\) 

b, `x^3 + 3x - 1 = 0` < Dề sai>

c, `x^2 + 4x = 21`

`<=> x^2 + 4x + 4 = 25`

`<=> (x + 2)^2  =25`

<=> \(\left[ \begin{array}{l}x + 2 = 5\\x + 2 = -5\end{array} \right.\) 

<=> \(\left[ \begin{array}{l}x=3\\x=-7\end{array} \right.\) 

d, `(2x - 1)^2 + (x + 3)^2 - 5(x + 7)(x - 7) = 0`

`<=> 4x^2 - 4x + 1 + x^2 + 6x + 9 - 5x^2 + 245 = 0`

`<=> 2x + 255 = 0`

`<=> 2x = -255`

`<=> x = -255/2`

e, `x^3 + x^2 + 1/3 x + 1/27 = 0`

`<=> x^3 + 3.x^2 . 1/3 + 3.x . (1/3)^2 + (1/3)^3 = 0`

`<=> (x + 1/3)^3 = 0`

`<=> x + 1/3 = 0`

`<=> x = -1/3`

2. 

a, Ta có : 

c1 : Tính đỡ nhóc nhưng làm dài

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

`<=> [(x + y - 2z)^2 - (x - y)^2] + [(z + x - 2y)^2 - (z - x)^2] + [(y + z - 2x)^2 - (y - z)^2] = 0`

`<=> (x + y - 2z - x + y)(x + y - 2z + x - y) + (z + x - 2y - z + x)(z + x - 2y + z - x) + (y + z - 2x - y + z)(y + z - 2x + y - z) = 0`

`<=> (2y - 2z)(2x - 2z) + (2x - 2y)(2z - 2y) + (2z - 2x)(2y - 2x) = 0`

`<=> 4.(y - z)(x - z) + 4.(x - y)(z - y) + 4.(z - x)(y - x) = 0`

`<=> (y - z)(x - z) + (x - y)(z - y) + (z - x)(y - x) =  0`

`<=> [(y - z)(x - z) + (x - y)(z - y)] + [(x - y)(z - y) + (z - x)(y - x)] + [(z - x)(y - x) + (y - z)(x - z)] = 0`

`<=> [(y - z)(x - z) - (x - y)(y - z)] + [(x - y)(y - z) - (x - y)(z - x)] + [(z - x)(y - x) - (z - x)(y - z)]= 0`

`<=> (y - z)(x - z - x + y) + (x - y)(z - y - z + x) + (z - x)(y - x - y + z) = 0`

`<=> (y - z)^2 + (x - y)^2 + (z - x)^2 = 0`

`<=> y - z = x - y = z - x = 0`

`<=> x = y = z`

c2 Ta có : 

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

`<=> (y + z - 2x)^2 + (z + x - 2y)^2 + (x + y - 2z)^2 - (x - y)^2 -  (y - z)^2 -  ( z - x)^2 = 0`

`<=> y^2 + z^2 + 4x^2 + 2yz - 4zx - 4xy + z^2 + x^2 + 4y^2 + 2zx - 4xy - 4yz + x^2 + y^2 + 4z^2 + 2xy - 4yz - 4zx - x^2 + 2xy - y^2 - y^2 + 2yz - z^2 - z^2 + 2zx - x^2 = 0`

`<=> 4x^2 + 4y^2 + 4z^2 - 4xy - 4yz - 4zx = 0`

`<=> 8x^2 + 8y^2 + 8z^2 - 8xy - 8yz - 8zx = 0`

`<=> (4x^2 - 8xy + 4y^2) + (4y^2 - 8yz + 4z^2) + (4z^2 - 8zx + 4x^2) = 0`

`<=> (2x - 2y)^2 + (2y - 2z)^2 + (2z - 2x)^2 = 0`

`<=> 2x - 2y = 2y - 2z = 2z - 2x = 0`

`<=> x - y = y - z = z - x = 0`

`<=> x = y = z`

b, Ta có :

`(x + y)(x^2 - xy + y^2) + (x -  y)(x^2 + xy + y^2) - 2x^3`

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

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

`= 0`

=> Biểu thức có giá trị không phụ thuộc vào biến

Giải thích các bước giải: