\(A=x^2+10y^2+2xy-6y+5\)

\(A=x^2+2xy+y^2+9y^2-6y+1+4\)

\(A=\left(x+y\right)^2+\left(3y+1\right)^2+4\)

Mà \(\hept{\begin{cases}\left(x+y\right)^2\ge0\\\left(3y+1\right)^2\ge0\\4>0\end{cases}}\)

=> A luôn dương với mọi x ; y

\(B=x-x^2-1\)

\(B=-\left(x^2-x+1\right)\)

\(B=-\left(x^2-x+\frac{1}{4}+\frac{3}{4}\right)\)

\(B=-\left[\left(x-\frac{1}{2}\right)^2+\frac{3}{4}\right]\)

\(B=-\left(x-\frac{1}{2}\right)^2-\frac{3}{4}\)

Mà \(\hept{\begin{cases}-\left(x-\frac{1}{2}\right)^2\le0\\-\frac{3}{4}< 0\end{cases}}\)

=> B luôn âm với mọi x

1. Ta có:

\(\frac{1}{x}+\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+...+\frac{1}{\left(x+2013\right)\left(x+2014\right)}\)

\(=\frac{1}{x}+\frac{1}{x}-\frac{1}{x+1}+\frac{1}{x+1}-\frac{1}{x+2}+...+\frac{1}{x+2013}-\frac{1}{x+2014}\)

\(=\frac{2}{x}-\frac{1}{x+2014}\)

\(=\frac{2\left(x+2014\right)}{x\left(x+2014\right)}-\frac{x}{x\left(x+2014\right)}\)

\(=\frac{2x+4028-x}{x\left(x+2014\right)}=\frac{x+4028}{x\left(x+2014\right)}\)

2a) ĐKXĐ: x \(\ne\)1 và x \(\ne\)-1

b) Ta có: A = \(\frac{x^2-2x+1}{x-1}+\frac{x^2+2x+1}{x+1}-3\)

A = \(\frac{\left(x-1\right)^2}{x-1}+\frac{\left(x+1\right)^2}{x+1}-3\)

A = \(x-1+x+1-3\)

A = \(2x-3\)

c) Với x = 3 => A = 2.3 - 3 = 3

c) Ta có: A = -2

=> 2x - 3 = -2

=> 2x = -2 + 3 = 1

=> x= 1/2

\(2x^2+5x+2\)

\(=2x^2+4x+x+2\)

\(=2x\left(x+2\right)+x+2\)

\(=\left(x+2\right)\left(2x+1\right)\)

2x² + 5x + 2

= 2x² + 4x + x + 2

= (2x² + 4x) + (x + 2)

= 2x (x + 2) + (x + 2)

= (2x + 1) (x + 2)

\(\frac{2}{2x+3}+\frac{5}{2x-3}-\frac{2x-33}{9-4x^2}\)

= \(\frac{2}{2x+3}+\frac{5}{2x-3}+\frac{2x-33}{4x^2-9}\)

= \(\frac{2\left(2x-3\right)}{\left(2x+3\right)\left(2x-3\right)}+\frac{5\left(2x+3\right)}{\left(2x-3\right)\left(2x+3\right)}+\frac{2x-33}{\left(2x-3\right)\left(2x+3\right)}\)

= \(\frac{4x-6+10x-15+2x-33}{\left(2x-3\right)\left(2x+3\right)}\)

= \(\frac{16x-54}{\left(2x-3\right)\left(2x+3\right)}\)

\(\frac{2}{2x+3}+\frac{5}{2x-3}-\frac{2x-33}{9-4x^2}\)\(=\frac{2}{2x+3}+\frac{5}{2x-3}+\frac{2x-33}{4x^2-9}\)

\(=\frac{2\left(2x-3\right)}{\left(2x+3\right)\left(2x-3\right)}+\frac{5\left(2x+3\right)}{\left(2x+3\right)\left(2x-3\right)}+\frac{2x-33}{\left(2x+3\right)\left(2x-3\right)}\)

\(=\frac{4x-6+10x+15+2x-33}{\left(2x+3\right)\left(2x-3\right)}=\frac{16x-24}{\left(2x+3\right)\left(2x-3\right)}=\frac{8\left(2x-3\right)}{\left(2x+3\right)\left(2x-3\right)}=\frac{8}{2x+3}\)