tìm x biết
\(\frac{\left(2019-x^2\right)+\left(2019-x\right)\left(x-2020\right)+\left(x-2020\right)^2}{\left(2019-x\right)^2-\left(2019-x\right)\left(x-2020\right)+\left(x-2020^2\right)}\) = \(\frac{19}{49}\)
Tính
\(\frac{1}{x\left(x+1\right)}+\frac{1}{\left(x+1\right)\left(x+2\right)}+\frac{1}{\left(x+2\right)\left(x+3\right)}+....+\frac{1}{\left(x+2019\right)\left(x+2020\right)}\)
Cho \(f\left(x\right)=\frac{x^3}{1-3x+3x^2}.\)Hãy tính giá trị của biểu thức sau:
\(B=f\left(\frac{1}{2020}\right)+f\left(\frac{2}{2020}\right)+..........+f\left(\frac{2018}{2020}\right)+f\left(\frac{2019}{2020}\right).\)
1. Giải phương trình nghiệm nguyên
a) \(x^2+4x+2018^{10}\)
b) \(x^2+4x+\left(y-1\right)^2=21\)
c) \(x^2+3\left(y-1\right)^2=2021\)
d) \(\left(3x-1\right)^{2020}-18\left(y-2\right)^{2019}=2019^{2020}\)
2. Tìm x,y ∈ Z
a) \(x^2-y^2+6y=56\)
b) \(x^2-4x+9y^2-6y=11\)
Ta có: \(f\left(2019\right)=2020=2019+1\)
\(f\left(2020\right)=2021=2020+1\)
Đặt \(h\left(x\right)=-x-1\)và \(g\left(x\right)=f\left(x\right)+h\left(x\right)\)
\(\Rightarrow\hept{\begin{cases}g\left(2019\right)=f\left(2019\right)+h\left(2019\right)=2020-2020=0\\g\left(2020\right)=f\left(2020\right)+h\left(2020\right)=2021-2021=0\end{cases}}\)
\(\Rightarrow x=2019;x=2020\)là nghiệm của đa thức g(x) mà g(x) là đa thức bậc 3 , hệ số \(x^3\)là số nguyên
\(\Rightarrow g\left(x\right)=a\left(x-2019\right)\left(x-2020\right)\left(x-x_0\right)\)(\(a\in\)Z*)
\(\Rightarrow f\left(x\right)=g\left(x\right)-h\left(x\right)\)
\(=a\left(x-2019\right)\left(x-2020\right)\left(x-x_0\right)+x+1\)
\(f\left(2021\right)=a\left(2021-2019\right)\left(2021-2020\right)\left(2021-x_0\right)+2021+1\)
\(=a.1.2\left(2021-x_0\right)+2022\)
\(f\left(2018\right)=a\left(2018-2019\right)\left(2018-2020\right)\left(2018-x_0\right)+2018+1\)
\(=a.1.2.\left(2018-x_0\right)+2019\)
\(\Rightarrow f\left(2021\right)-f\left(2018\right)=a.1.2\left(2021-2018\right)+3\)
\(=6a+3\)
Làm nốt
1,tìm x biết:
\(\left|x-2019\right|^{2019}+\left|x-2020\right|^{2020}=1\)
Tìm GTLN của A:
A=\(\left|x-2019\right|+\left|x-2020\right|\)
Tìm GTNN của A:
A=\(\left|x-2019\right|+\left|x-2020\right|\)
Bài 1
Cho \(\left\{{}\begin{matrix}a+b+c=0\\ab+ba+ca=0\end{matrix}\right.\)
Tính \(A=\left(a-1\right)^{2019}+\left(b-1\right)^{2020}+\left(c-1\right)^{2021}\)
Bài 2 Tìm a,b,c ∈Z sao cho
\(\left(x+b\right)\left(x+c\right)=\left(x+a\right)\left(x-4\right)-7\)
Bài 3 Tìm a,b,c sao cho
\(x^3+ax^{2\:}+bx+c=\left(x+a\right)\left(x+b\right)\left(x+c\right)\)