Bài 1 : 

a, Xét tam giác ABC, theo định lí Ta lét ta có 

\(\frac{EC}{AE}=\frac{DC}{BD}\Rightarrow AE=x=\frac{EC.BD}{DC}=\frac{12.3}{5}=\frac{36}{5}\)cm 

b, Vì \(MN\perp HI;HK\perp HI\Rightarrow\)HK // MN 

Theo hệ quả Ta lét \(\frac{MI}{HI}=\frac{MN}{HK}\Rightarrow MN=y=\frac{MI.HK}{HI}=\frac{2.8}{6}=\frac{8}{3}\)

Theo định lí Pytago tam giác MNI vuông tại M 

\(IN=\sqrt{MI^2+MN^2}=\sqrt{4+\frac{64}{9}}=\frac{10}{3}\)

\(12x-9-4x^2\)\(=-\left(4x^2-12x+9\right)\)\(=-\left[\left(2x\right)^2-2.2x.3+3^2\right]\)\(=-\left(2x-3\right)^2\)

2/5.xy(x²y-5x+10y)

=2/5.xy.x²y-2/5.xy.5x+2/5.xy.10y

=2/5.x³y²-2x²y+4xy²

\(\frac{1}{1-x}+\frac{1}{1+x}+\frac{2}{1+x^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{1+x+1-x}{\left(1-x\right)\left(1+x\right)}+\frac{2}{1+x^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{2}{1-x^2}+\frac{2}{1+x^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{2\left(1+x^2\right)+2\left(1-x^2\right)}{\left(1-x^2\right)\left(1+x^2\right)}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{2+2x^2+2-2x^2}{1-\left(x^2\right)^2}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{4}{1-x^4}+\frac{4}{1+x^4}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{4\left(1+x^4\right)+4\left(1-x^4\right)}{\left(1-x^4\right)\left(1+x^4\right)}+\frac{8}{1+x^8}+\frac{16}{1+x^8}\)

\(=\frac{4+4x^4+4-4x^4}{1-\left(x^4\right)^2}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

\(=\frac{8}{1-x^8}+\frac{8}{1+x^8}+\frac{16}{1+x^{16}}\)

Cứ tiếp tục vậy cho đến KQ là \(\frac{32}{1-x^{32}}\)