2. Ta có: P = 2x² + y² - 4x - 4y + 10

P = 2(x² - 2x + 1) + (y² - 4y + 4) + 4

P = 2(x - 1)² + (y - 2)² + 4 \(\ge\)4 \(\forall\)x;y

=> P luôn dương với mọi biến x;y

3 Ta có:

(2n + 1)(n² - 3n - 1) - 2n³ + 1

= 2n³ - 6n² - 2n + n² - 3n - 1 - 2n³ + 1

= -5n² - 5n = -5n(n + 1) \(⋮\)5 \(\forall\)n \(\in\)Z

1×2=2

a) A = x(y - z) + 2(z - y) = x(y - z) - 2(y - z) = (x - 2)(y - z) = (2 - 2)(1,007 - (-0,006)] = 0

b) B = 2x(y - z) + (z - y)(x + t) = 2x(y - z)  - (y - z)(x + t) = (2x - x - t)(y - z) = (x - t)(y - z) = [18,3 - (-31,7)](24,6 - 10,6) = 50.14 = 700

c) C = (x - y)(y + z) + y(y - x) = (x - y)(y + z) - y(x - y) = (x - y)(y + z - y) = (x - y).z = (0,86 - 0,26).1,5 = 0,6.1,5 = 0,9

\(b)4x\left(x-2014\right)-\left(x-2014\right)=0\)

\(\left(4x-1\right)\left(x-2014\right)=0\)

\(\Leftrightarrow TH1:4x-1=0\)

\(4x=1\)

\(x=\frac{1}{4}\)

\(TH2:x-2014=0\)

\(x=2014\)

Vậy \(x\in\left\{\frac{1}{4};2014\right\}\)

\(b,4x\left(x-2014\right)-x+2014=0\)

\(\Leftrightarrow\left(x-2014\right)\left(4x-1\right)=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=2014\\x=\frac{1}{4}\end{cases}}\)

\(c,\left(x+1\right)^2=x+1\)

\(\Leftrightarrow\left(x+1\right)x=0\)

\(\Leftrightarrow\orbr{\begin{cases}x=0\\x=-1\end{cases}}\)

\(1.=5xy\left(x-2y\right)\)

\(2.=\left(5-y\right)\left(x-y\right)\)

\(3.=y\left(x-z\right)-7\left(x-z\right)=\left(y-7\right)\left(x-z\right)\)

\(5.=2x\left(3y-7z\right)-6y\left(3y-7z\right)=\left(2x-6y\right)\left(3y-7x\right)\)

\(4.=27x^2\left(y-1\right)+9x^3\left(y-1\right)=9x^2\left(3+x\right)\left(y-1\right)\)

k

\(30xy-16x^2-9y^2\)

\(=-\left(16x^2-24xy+9y^2\right)+6xy\)

\(=-\left(4x-3y\right)^2+6xy\)

\(=-\left[\left(4x-3y\right)^2-6xy\right]\)

\(=-\left(4x-3y-\sqrt{6xy}\right)\left(4x-4y+\sqrt{6xy}\right)\)

Ta có: \(2005\equiv-1\left(mod2006\right)\)

\(\Rightarrow2005^{2007}\equiv-1\left(mod2006\right)\)

Lại có: \(2007=1\left(mod2006\right)\)

\(\Rightarrow2007^{2005}\equiv1\left(mod2006\right)\)

\(\Rightarrow2005^{2007}+2007^{2005}\equiv0\left(mod2006\right)\)

Vậy \(2005^{2007}+2007^{2005}⋮2006\left(đpcm\right)\)

mod là gì