cho phuong trinh \(x^2-2x+m^2-2m+1=0\left(1\right)\) voi m la tham so

a/ Giai phuong trinh (1) khi m=\(\sqrt{2}\)

b/Chung minh rang neu phuong trinh (1) co 2 nghiem \(x_1,x_2\) thi \(\left|x_2-x_1\right|\le2\)

cho abc=1 tim GTLN M=\(\frac{1}{a^2+2b^2+3}+\frac{1}{b^2+2c^2+3}+\frac{1}{c^2+2a^2+3}\)

rut gon H=\(\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)

chp phuong trinh bac hai \(x^2-4x+4m-m^2=0\)

a/chung minh phuong trinh luon co nghiem voi moi gia tri cua m

b/ goi \(x_1,x_2\) la nghiem cua phuong trinh da cho . Hay tinh gia tri cua m sao cho \(x_2=x_1^2-5x_1\)

giai pt \(\frac{1}{x^2+5x+6}+\frac{1}{x^2+7x+12}+\frac{1}{x^2+9x+20}+\frac{1}{x^2+11x+30}=\frac{1}{8}\)

gia su cac so a,b thoa \(\left\{{}\begin{matrix}a^3-3ab^2=233\\b^3-3a^2b=2010\end{matrix}\right.\) Tinh P=\(a^2+b^2\)

cho 13,6g Ca va CaO tac dung hoan toan voi nuoc thi thu duoc 4,48l khi \(H_2\) (DKTC)

a/ tinh khoi luong moi chat co trong hon hop ban dau

b/ tinh khoi luong bazo tao thanh sau phan ung

cho a , b, c la cac so thuc duong thoa man he thuc a+b+c=6abc

Chung minh rang \(\dfrac{bc}{a^3\left(c+2b\right)}+\dfrac{ac}{b^3\left(a+2c\right)}+\dfrac{ab}{c^3\left(b+2a\right)}\ge2\)

tim gia tri nho nhat cua bieu thuc P=\(\left(1+x\right)\left(1+\dfrac{1}{y}\right)+\left(1+y\right)\left(1+\dfrac{1}{x}\right)\) trong do x,y la cac so duong thoa man \(x^2+y^2=1\)