Question

[LÂM 29]

giúp em với em đang cần gấp lắm mọi người ơi huhu

Answer 1

[Nguyễn Việt Lâm]

1.

a.

\(n^2+7n+1=k^2\Rightarrow4n^2+28n+4=4k^2\)

\(\Leftrightarrow\left(2n+7\right)^2-45=\left(2k\right)^2\)

\(\Leftrightarrow\left(2n-2k+7\right)\left(2n+2k+7\right)=45\)

Phương trình ước số cơ bản

b.

\(a^3b^3+b^3-3ab^2=-1\)

\(\Leftrightarrow a^3+1-\dfrac{3a}{b}=-\dfrac{1}{b^3}\)

\(\Leftrightarrow a^3+\dfrac{1}{b^3}+1-\dfrac{3a}{b}=0\)

Đặt \(\left(a;\dfrac{1}{b}\right)=\left(x;y\right)\Rightarrow x^3+y^3+1-3xy=0\)

\(\Leftrightarrow\left(x+y\right)^3+1-3xy\left(x+y\right)-3xy=0\)

\(\Leftrightarrow\left(x+y+1\right)\left(x^2+y^2+1-xy-x-y\right)=0\)

\(\Leftrightarrow x+y+1=0\)

\(\Rightarrow P=a+\dfrac{1}{b}=x+y=-1\)

Answer 2

[Nguyễn Việt Lâm]

2.

a.

 \(a+b+\dfrac{1}{a}+\dfrac{1}{b}=\left(\dfrac{a}{4}+\dfrac{1}{a}\right)+\left(\dfrac{b}{4}+\dfrac{1}{b}\right)+\dfrac{3}{4}\left(a+b\right)\)

\(\ge2\sqrt{\dfrac{a}{4a}}+2\sqrt{\dfrac{b}{4b}}+\dfrac{3}{4}.4=5\) (đpcm)

Dấu "=" xảy ra khi \(a=b=2\)