Question

[Kiriya Aoi]

a) Cho a² + b² + c² = ab + bc + ca, chứng minh a = b = c.

b) Tìm a,b,c thỏa mãn đẳng thức : a² - 2a + b² + 4b + 4c² - 4c + 6 = 0

Answer 1

[Trần Quốc Lộc]

\(a^2-2a+b^2+4b+4c^2-4c+6=0\\ \Leftrightarrow\left(a^2-2a+1\right)+\left(b^2+4b+4\right)+\left(4c^2-4c+1\right)=0\\ \Leftrightarrow\left(a+1\right)^2+\left(b+2\right)^2+\left(2c-1\right)^2=0\\ \Leftrightarrow\left\{{}\begin{matrix}\left(a+1\right)^2=0\\\left(b+2\right)^2=0\\\left(2c-1\right)^2=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a+1=0\\b+2=0\\2c-1=0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}a=-1\\b=-2\\c=\dfrac{1}{2}\end{matrix}\right.\)

Vậy \(\left\{a;b;c\right\}=\left\{-1;-2;\dfrac{1}{2}\right\}\)

Answer 2

[hattori heiji]

a nhân 2 vào 2 vế ta có

2a²+2b²+2c²=2ab +2bc+2ca

=> 2a²+2b²+2c²-2ab-2bc-2ca=0

=>(a²-2ab+b²)+(b²-2bc+c²)+(c²-2ca+a²)=0

=>(a-b)²+(b-c)²+(c-a)²=0

=>(a-b)=(b-c)=(c-a)=0

=>a-b=0 =>a=b (1)

b-c=0=>b=c (2)

từ (1) và (2)

=>a=b=c (đpcm)