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Tại hạ đã biết là thánh học lớp 8 
Cao :\_________________________________/
a)\(\left(ab+bc+ca\right)^2=a^2b^2+b^2c^2+c^2a^2\)\(+2\left(ab^2c+abc^2+a^2bc\right)\)
=\(a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)\)
=\(a^2b^2+b^{2^2}c^2+c^2a^2+2abc.0\)
=\(a^2b^2+b^2c^2+c^2a^2\)
b) \(a+b+c=0\)=>\(\left(a+b+c\right)^2=0\)
<=>\(a^2+b^2+c^2+2\left(ab+bc+xa\right)=0\)
<=>\(a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
=>\(\left(a^2+b^2+c^2\right)^2=4\left(ab+bc+ca\right)^2\)
<=>\(a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)\)\(=4\left(ab+bc+ca\right)^2\)
Do \(\left(ab+bc+ca\right)^2=a^2b^2+b^2c^2+c^2a^2\)
=>\(a^4+b^4+c^4+2\left(ab+bc+ca\right)^2\)\(=4\left(ab+bc+ca\right)^2\)
=>\(a^4+b^4+c^4=2\left(ab+bc+ca\right)^2\)
Ta có : a + b + c = 0
( a + b + c )\(^2\) = 0
\(a^2+b^2+c^2+2ab+2bc+2ca=0\)
Nên : \(a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
\(\left(a^2+b^2+c^2\right)^2=4\left(ab+bc+ca\right)^2\)
\(a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2=4\left(ab+bc+ca\right)^2\)
\(a^4+b^4+c^4+2a^2b^2+2b^2c^2+2c^2a^2=4\left(a^2b^2+b^2c^2+c^2a^2+2ab^2c+2abc^2+2a^2bc\right)\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2+8ab^2c+8abc^2+8a^2bc\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2+8abc\left(b+c+a\right)\)
\(a^4+b^4+c^4=2a^2b^2+2b^2c^2+2c^2a^2\)
Lại có : \(2\left(ab+bc+ca\right)^2\)
\(=2\left(a^2b^2+b^2c^2+c^2a^2+2ab^2c+2abc^2+2a^2bc\right)\)
\(=2a^2b^2+2b^2c^2+2c^2a^2+4ab^2c+4abc^2+4a^2bc\)
\(=2a^2b^2+2b^2c^2+2c^2a^2+4abc\left(b+c+a\right)\)
\(=2a^2b^2+2b^2c^2+2c^2a^2\)
Vì : \(2a^2b^2+2b^2c^2+2c^2a^2=2a^2b^2+2b^2c^2=2c^2a^2\)
Vậy \(a^4+b^4+c^4=2\left(ab+bc+ca\right)^2\)
\(\left(ab+bc+ac\right)^2=a^2b^2+b^2c^2+c^2a^2\\ \Leftrightarrow a^2b^2+b^2c^2+a^2c^2+2\left(ab^2c+abc^2+a^2bc\right)=a^2b^2+b^2c^2+c^2a^2\\ \Leftrightarrow2\left(ab^2c+abc^2+a^2bc\right)=0\\ \Leftrightarrow abc\left(a+b+c\right)=0\left(đpcm;a+b+c=0\right)\)
Sửa lại đề: \(a+b+c=0\)
a) Ta có:
\(A=a^4+b^4+c^4=(a^2+b^2+c^2)^2-2(a^2b^2+b^2c^2+c^2a^2)\)
\(=[(a+b+c)^2-2(ab+bc+ac)]^2-2(a^2b^2+b^2c^2+c^2a^2)\)
\(=4(ab+bc+ac)^2-2(a^2b^2+b^2c^2+c^2a^2)\)
\(=4(ab+bc+ac)^2-2(a^2b^2+b^2c^2+c^2a^2)-4abc(a+b+c)\)
(do \(a+b+c=0\))
\(A=4(ab+bc+ac)^2-2[a^2b^2+b^2c^2+c^2a^2+2abc(a+b+c)]\)
\(=4(ab+bc+ac)^2-2(ab+bc+ac)^=2(ab+bc+ac)^2\)
Ta có đpcm
b) Ta có:
\(\frac{(a^2+b^2+c^2)^2}{2}=\frac{[(a+b+c)^2-2(ab+bc+ac)]^2}{2}=\frac{[-2(ab+bc+ac)]^2}{2}=2(ab+bc+ac)^2\)
Kết hợp với kết quả phần a ta có đpcm.
Bạn ơi cái chỗ
= 4(ab+bc+ca)^2 - 2(ab+bc+ca)= 2(ab+bc+ca)^2
thì phải là như thế này chứ
= 4(ab+bc+ca)^2 - 2(ab+bc+ca)^2= 2(ab+bc+ca)^2
Đây là ý mình còn nếu ko phải mong bạn bỏ qua và giải thích cho mình nhé!!
\(\left(a+b+c\right)^2+12=4\left(a+b+c\right)\)\(+2\left(ab+bc+ac\right)=0\)
\(\Rightarrow a^2+b^2+c^2+2ab+2bc+2ac+12-4\left(a+b+c\right)-2\left(ab+bc+ac\right)=0\)
\(\Rightarrow a^2+b^2+c^2+2\left(ab+ac+bc\right)-2\left(ab+bc+ac\right)-4\left(a+b+c\right)+12=0\)
\(\Rightarrow a^2+b^2+c^2-4a-4b-4c+12=0\)
\(\Rightarrow\left(a^2-4a+4\right)+\left(b^2-4b+4\right)+\left(c^2-4c+4\right)=0\)
\(\Rightarrow\left(a-2\right)^2+\left(b-2\right)^2+\left(c-2\right)^2=0\)
Ta co: \(\left(a-2\right)^2\ge0\forall a\)
\(\left(b-2\right)^2\ge0\forall b\)
\(\left(c-2\right)^2\ge0\forall c\)
\(\Rightarrow\left(a-2\right)^2+\left(b-2\right)^2+\left(c-2\right)^2=0\Leftrightarrow\hept{\begin{cases}\left(a-2\right)^2=0\\\left(b-2\right)^2=0\\\left(c-2\right)^2=0\end{cases}\Leftrightarrow\hept{\begin{cases}a-2=0\\b-2=0\\c-2=0\end{cases}\Leftrightarrow}a=b=c=2}\left(\right)\)
(đpcm)
Mình nghĩ thế này nhé bạn!
(a + b + c )2 + 12 = 4 (a + b +c ) + 2(ab + bc +ac)
\(\Leftrightarrow\)a2 + b2 + c2 + 2ab + 2bc + 2ac + 12 = 4a + 4b + 4c + 2ab + 2ac + 2bc
\(\Leftrightarrow\) a2 + b2 + c2 - 4a - 4b -4c +12 = 0
\(\Leftrightarrow\)a2 - 4a + 4 + b2 - 4b + 4 + c2 - 4c + 4 =0
\(\Leftrightarrow\)( a -2 )2 + (b-2)2 + (c-2)2 = 0
ta có (a-2 )2 \(\ge0\forall a\)
(b - 2 )2 \(\ge0\forall b\)
(c - 2 )2 \(\ge0\forall c\)
mà (a-2)2 + (b-2)2 + (c-2)2 = 0
\(\Rightarrow\hept{\begin{cases}\left(a-2\right)^2=0\\\left(b-2\right)^2=0\\\left(c-2\right)^2=0\end{cases}}\Leftrightarrow\hept{\begin{cases}a-2=0\\b-2=0\\c-2=0\end{cases}}\)
\(\Leftrightarrow\hept{\begin{cases}a=2\\b=2\\c=2\end{cases}\left(đpcm\right)}\)
vậy................... khi a=b = c =2
#mã mã#
1)Áp dụng Bđt Am-Gm \(\frac{a}{b}+\frac{b}{a}\ge2\sqrt{\frac{a}{b}\cdot\frac{b}{a}}=2\)
2)Áp dụng Am-Gm \(a^2+b^2\ge2\sqrt{a^2b^2}=2ab;b^2+c^2\ge2bc;a^2+c^2\ge2ca\)
\(\Rightarrow2\left(a^2+b^2+c^2\right)\ge2\left(ab+bc+ca\right)\)
=>ĐPcm
3)(a+b+c)2\(\ge\)3(ab+bc+ca)
=>a2+b2+c2+2ab+2bc+2ca\(\ge\)3ab+3bc+3ca
=>a2+b2+c2-ab-bc-ca\(\ge\)0
=>2a2+2b2+2c2-2ab-2bc-2ca\(\ge\)0
=>(a2-2ab+b2)+(b2-2bc+c2)+(c2-2ac+a2)\(\ge\)0
=>(a-b)2+(b-c)2+(c-a)2\(\ge\)0
4)đề đúng \(\frac{1}{a}+\frac{1}{b}\ge\frac{4}{a+b}\)
\(\Leftrightarrow\frac{a+b}{ab}\ge\frac{4}{a+b}\)
\(\Leftrightarrow\left(a+b\right)^2\ge4ab\)
\(\Leftrightarrow a^2+2ab+b^2-4ab\ge0\)
\(\Leftrightarrow\left(a-b\right)^2\ge0\)
lần sau đăng bài bạn nhớ đăng đúng đề nhé
sửa đề: \(\left(a+b+c\right)^2+12=4\left(a+b+c\right)+2\left(ab+bc+ac\right)
\)
=> \(a^2+b^2+c^2+2ab+2bc+2ac+12-4a-4b-4c-2ab-2bc-2ac=0\)
=> \(a^2+b^2+c^2-4a-4b-4c+12=0\)
=>\(\left(a-2\right)^2+\left(b-2\right)^2+\left(c-2\right)^2=0\)
=> a=b=c=2
Câu a/ Thì chứng minh ở dưới rồi nhé e
b/ Ta cần chứng minh
\(2\left(a^2b^2+b^2c^2+c^2a^2\right)=2\left(ab+bc+ca\right)^2\)
\(\Leftrightarrow a^2b^2+b^2c^2+c^2a^2=a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\)
\(\Leftrightarrow2abc\left(a+b+c\right)=0\)(đúng)
=> ĐPCM
c/ Ta có
\(\frac{\left(a^2+b^2+c^2\right)^2}{2}=\frac{a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)}{2}=a^4+b^4+c^4\)
Cái này là áp dụng câu a vô nhé e
Ta có :
\(\left(a+b+c\right)=0\)
\(\Leftrightarrow\left(a+b+c\right)^2=0\)
\(\Leftrightarrow a^2+b^2+c^2+2\left(ab+bc+ca\right)=0\)
\(\Leftrightarrow a^2+b^2+c^2=-2\left(ab+bc+ca\right)\)
\(\Leftrightarrow\left(a^2+b^2+c^2\right)^2=\left[-2\left(ab+bc+ca\right)\right]^2\)
\(\Leftrightarrow a^4+b^4+c^4+2\left(a^2b^2+b^2c^2+c^2a^2\right)=4\left(a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\right)\left(1\right)\)
\(\Leftrightarrow a^4+b^4+c^4=4\left(a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right)\right)-2\left(a^2b^2+b^2c^2+c^2a^2\right)\)
\(\Leftrightarrow a^4+b^4+c^4=2\left(a^2b^2+b^2c^2+c^2a^2\right)\left(2\right)\) (vì \(a+b+c=0\))
\(\left(1\right)+\left(2\right)\Rightarrow2\left(a^4+b^4+c^4\right)=4\left(a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2\right)\)
\(\Rightarrow\left(a^4+b^4+c^4\right)=2\left(ab+bc+ca\right)^2\)
\(\Rightarrow dpcm\)