Question

[Nguyễn Lê Bảo Quyên]

1. K= ​\(\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{1}{a-\sqrt{a}}\right):\left(\frac{1}{\sqrt{a}+1}+\frac{2}{a-1}\right)\)

a) rút gọn k

b) tính k khi a=\(3+2\sqrt{2}\)

2. giải hpt

\(\hept{\begin{cases}x-y=3\\x^3-y^3=9\end{cases}}\)

Answer 1

[Bùi Thế Hào]

1/ a/  \(K=\left(\frac{\sqrt{a}}{\sqrt{a}-1}-\frac{1}{\sqrt{a}\left(\sqrt{a}-1\right)}\right):\left(\frac{1}{\sqrt{a}+1}+\frac{2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\right)\)

\(K=\frac{a-1}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{\sqrt{a}-1+2}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\)

\(K=\frac{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}{\sqrt{a}\left(\sqrt{a}-1\right)}:\frac{\sqrt{a}+1}{\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)}\)

\(K=\frac{\sqrt{a}+1}{\sqrt{a}}:\frac{1}{\sqrt{a}-1}=\frac{\sqrt{a}+1}{\sqrt{a}}.\left(\sqrt{a}-1\right)\)

\(K=\frac{a-1}{\sqrt{a}}\)

b/ Với \(a=3+2\sqrt{2}\) => \(K=\frac{a-1}{\sqrt{a}}=\frac{3+2\sqrt{2}-1}{\sqrt{3+2\sqrt{2}}}=\frac{2+2\sqrt{2}}{\sqrt{2+2\sqrt{2}+1}}=\frac{2\left(\sqrt{2}+1\right)}{\sqrt{\left(\sqrt{2}+1\right)^2}}=\frac{2\left(\sqrt{2}+1\right)}{\left(\sqrt{2}+1\right)}\)

=> \(K=2\)

Answer 2

[Bùi Thế Hào]

2/ Ta có: x³-y³=x-y)(x²+xy+y²)=(x-y)(x²-2xy+y²+3xy)=(x-y)[(x-y)²+3xy]=9

Thay x-y=3 vào ta được: 3(9+3xy)=9

<=> 3+xy=1  => xy=-2

Ta có hệ PT: \(\hept{\begin{cases}x-y=3\\xy=-2\end{cases}}\)=> \(\hept{\begin{cases}x=y+3\\xy=-2\end{cases}}\)

=> y(y+3)+2=0

<=> y²+3y+2=0

<=> y²+y+2y+2=0  <=> y(y+1)+2(y+1)=0  <=> (y+1)(y+2)=0

=> y₁=-1  => x₁=2

y₂=-2 => x₂=1

Đáp số: Các cặp x,y là: (2; -1) và (1; -2)