Áp dụng BĐT Cosi cho 3 số x,y,z dương ta có:

\(x+y\ge2\sqrt{xy};y+z\ge2\sqrt{yz};z+x\ge2\sqrt{zx}\)

Nhân các BĐT vế theo vế ta được:

\(\left(x+y\right)\left(y+z\right)\left(z+x\right)\ge2\sqrt{xy}.2\sqrt{yz}.2\sqrt{zx}=8\sqrt{x^2y^2z^2}=8xyz\)

Dấu "=" xảy ra khi x = y = z

<=> x-y=y-z=z-x=0

<=>(x-y)²+(y-z)²+(z-x)²=0

<=>x²-2xy+y²+y²-2yz+z²+z²-2zx+x²=0

<=>2x²+2y²+2z²-2xy-2yz-2zx=0

<=>x²+y²+z²-xy-yz-zx=0

<=>(x+y+z)(x²+y²+z²-xy-yz-zx)=0 (vì x,y,z>0 nên x+y+z>0)

<=>x³+y³+z³-3xyz=0

<=>x³+y³+z³=3xyz (đpcm)

a) Ta có \(\left(x-y\right)^2=x^2-2xy+y^2=\left(x^2+2xy+y^2\right)-4xy\)

\(=\left(x+y\right)^2-4xy=9^2-4.14=25\)

Vậy nên \(\orbr{\begin{cases}x-y=5\\x-y=-5\end{cases}}\)

b) \(x^2+y^2=\left(x^2+2xy+y^2\right)-2xy=\left(x+y\right)^2-2xy\)

\(=9^2-2.14=53\)

c) \(x^3+y^3=\left(x+y\right)\left(x^2-xy+y^2\right)=\left(x+y\right)\left[\left(x+y\right)^2-3xy\right]\)

\(=9.\left(9^2-3.14\right)=351\)

x² + x  + 1 

= x ² +2x +1 - x 

= (x + 1 )² - \(\sqrt{x}\)²

= ( x  + 1 - \(\sqrt{x}\) ) (x + 1 + \(\sqrt{x}\))

\(x^2+x+1=\left[x^2+2.\frac{1}{2}x+\left(\frac{1}{2}\right)^2\right]+\left(\frac{\sqrt{3}}{2}\right)^2\)

\(=\left(x^2+\frac{1}{2}\right)-\left(\frac{\sqrt{3}}{2}\right)^2\)

\(=\left(x^2+\frac{1}{2}-\frac{\sqrt{3}}{2}\right)\left(x^2+\frac{1}{2}+\frac{\sqrt{3}}{2}\right)\)

\(=\left(x^2+\frac{1-\sqrt{3}}{2}\right)\left(x^2+\frac{1+\sqrt{3}}{2}\right)\)

Tham khảo nhé~

Ta có: 2005²⁰⁰⁷ + 2007²⁰⁰⁵ = (2005²⁰⁰⁷ + 1²⁰⁰⁷ ) + (2007²⁰⁰⁵ - 1²⁰⁰⁵ )

Vì \(2005^{2007}+1^{2007}\)luôn chia hết cho \(2005+1=2006\left(1\right)\)

    \(2007^{2005}-1^{2005}\)luôn chia hết cho \(2007-1=2006\left(2\right)\)

\(\left(1\right)\left(2\right)\Rightarrow\left(2005^{2007}+1^{2007}\right)+\left(2007^{2005}-1^{2005}\right)⋮2006\)

                 \(\Rightarrow2005^{2007}+2007^{2005}⋮2006\)

Vậy \(2005^{2007}+2007^{2005}⋮2006\)

127² + 146.127 + 73²

= 127² + 2 . 73 .127 + 73²

= (127 + 73 ) ²

= 200 ²

Ta có : \(2(x-y)(x+y)+(x+y)^2+(x-y)^2\)

      \(=2(x^2-y^2)+2(x^2+y^2)\)

       \(=2x^2+2x^2+2y^2-2y^2=4x^2\)

Chúc bạn học tốt

\(2\left(x-y\right)\left(x+y\right)+\left(x+y\right)^2+\left(x-y\right)^2\)

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

\(=2x^2-2y^2+x^2+2xy+y^2+x^2-2xy+y^2\)

\(=4x^2\)

Chúc bạn học tốt!

a, \(A=-1^2+2^2-3^2+4^2-...-99^2+100^2\)

\(=-\left(1^2-2^2+3^2-4^2+...+99^2-100^2\right)\)

\(=-\left[\left(1+2\right)\left(1-2\right)+\left(3+4\right)\left(3-4\right)+...+\left(99+100\right)\left(99-100\right)\right]\)

\(=-\left(-3-7-...-199\right)\)

\(=3+7+...+199\)

\(=\frac{\left(199+3\right).50}{2}=5050\)