Question

Cho x+y=9 và xy=14. Không tính x,y. Hãy tính giá trị biểu thức

1, A=x-y

2, B=x^2 + y^2

3, C=x^3 + y^3

4, D= x^4 + y^4

5, E= x^5 + y^5

Answer 1

1) \(x+y=9\Rightarrow\left(x+y\right)^2=81\Rightarrow x^2+2xy+y^2=81\Rightarrow\left(x-y\right)^2=81-4xy=81-4.14=25\Rightarrow\left|x-y\right|=5\Rightarrow\left[{}\begin{matrix}x-y=5\\x-y=-5\end{matrix}\right.\)

2) \(x^2+2xy+y^2=81\Rightarrow x^2+y^2=81-2xy=81-2.14=53\)

3) \(C=x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=9\left(53-14\right)=351\)

4) \(\left(x^2+y^2\right)^2=2809\Rightarrow x^4+2x^2y^2+y^4=2809\Rightarrow x^4+y^4=2809-2x^2y^2=2809-2.14^2=2417\)

5) \(\left(x^2+y^2\right)\left(x^3+y^3\right)=18603\Rightarrow x^5+y^5+x^2y^2\left(x+y\right)=18603\Rightarrow x^5+y^5=18603-x^2y^2\left(x+y\right)=18603-14^2.9=16839\)