Bài 2:

a) \(A=\dfrac{a^2}{bc}+\dfrac{b^2}{ca}+\dfrac{c^2}{ab}\)

\(A=\dfrac{a^3}{abc}+\dfrac{b^3}{abc}+\dfrac{c^3}{abc}\)

\(A=\dfrac{1}{abc}\left(a^3+b^3+c^3\right)\)

\(A=\dfrac{1}{abc}\left[\left(a+b\right)^3-3ab\left(a+b\right)+c^3\right]\)

Vì \(a+b+c=0\)

Nên a + b = -c (1)

Thay (1) vào A, ta được:

\(A=\dfrac{1}{abc}\left[\left(-c\right)^3-3ab\left(-c\right)+c^3\right]\)

\(A=\dfrac{1}{abc}.3abc\)

\(A=3\)

b) \(B=\dfrac{a^2}{a^2-b^2-c^2}+\dfrac{b^2}{b^2-c^2-a^2}+\dfrac{c^2}{c^2-a^2-b^2}\)

\(B=\dfrac{a^2}{a^2-\left(b^2+c^2\right)}+\dfrac{b^2}{b^2-\left(c^2+a^2\right)}+\dfrac{c^2}{c^2-\left(a^2+b^2\right)}\)

Vì \(a+b+c=0\)

Nên b + c = -a

=> ( b + c )² = (-a)²

=> b² + c² + 2bc = a²

=> b² + c² = a² - 2bc (1)

Tương tự ta có: c² + a² = b² - 2ac (2)

a² + b² = c - 2ab (3)

Thay (1), (2) và (3) vào B, ta được:

\(B=\dfrac{a^2}{a^2-\left(a^2-2bc\right)}+\dfrac{b^2}{b^2-\left(b^2-2ac\right)}+\dfrac{c^2}{c^2-\left(c^2-2ab\right)}\)

\(B=\dfrac{a^2}{a^2-a^2+2bc}+\dfrac{b^2}{b^2-b^2+2ac}+\dfrac{c^2}{c^2-c^2+2ab}\)

\(B=\dfrac{a^2}{2bc}+\dfrac{b^2}{2ac}+\dfrac{c^2}{2ab}\)

\(B=\dfrac{a^3}{2abc}+\dfrac{b^3}{2abc}+\dfrac{c^3}{2abc}\)

\(B=\dfrac{1}{2abc}\left(a^3+b^3+c^3\right)\)

Mà \(a^3+b^3+c^3=3abc\) ( câu a )

\(\Rightarrow B=\dfrac{1}{2abc}.3abc\)

\(\Rightarrow B=\dfrac{3}{2}\)

Bài 1:

a) GT: abc = 2

\(M=\dfrac{a}{ab+a+2}+\dfrac{b}{bc+b+1}+\dfrac{2c}{ac+2c+2}\)

\(M=\dfrac{a}{ab+a+abc}+\dfrac{b}{bc+b+1}+\dfrac{2cb}{abc+2cb+2b}\)

\(M=\dfrac{a}{a\left(b+1+bc\right)}+\dfrac{b}{bc+b+1}+\dfrac{2cb}{2+2cb+2b}\)

\(M=\dfrac{1}{bc+b+1}+\dfrac{b}{bc+b+1}+\dfrac{2cb}{2\left(1+cb+b\right)}\)

\(M=\dfrac{1}{bc+b+1}+\dfrac{b}{bc+b+1}+\dfrac{bc}{bc+b+1}\)

\(M=\dfrac{1+b+bc}{bc+b+1}\)

\(M=1\)

b) GT: abc = 1

\(N=\dfrac{a}{ab+a+1}+\dfrac{b}{bc+b+1}+\dfrac{c}{ac+c+1}\)

\(N=\dfrac{a}{ab+a+abc}+\dfrac{b}{bc+b+1}+\dfrac{cb}{b\left(ac+c+1\right)}\)

\(N=\dfrac{a}{a\left(b+1+bc\right)}+\dfrac{b}{bc+b+1}+\dfrac{bc}{abc+bc+b}\)

\(N=\dfrac{1}{bc+b+1}+\dfrac{b}{bc+b+1}+\dfrac{bc}{bc+b+1}\)

\(N=\dfrac{1+b+bc}{bc+b+1}\)

\(N=1\)

Bài 2:

Bài 1:

\(a^2+b^2+c^2=14\Rightarrow\left(a+b+c\right)^2-2ab-2bc-2ac=14\)\(\Leftrightarrow-2\left(ab+bc+ac\right)=14\Rightarrow ab+bc+ac=-7\)\(\Rightarrow\left(ab+bc+ac\right)^2=49\)

\(\Leftrightarrow a^2b^2+b^2c^2+c^2a^2+2a^2bc+2ab^2c+2abc^2=49\)\(\Leftrightarrow a^2b^2+b^2c^2+a^2c^2+2abc\left(a+b+c\right)=49\)

\(\Rightarrow a^2b^2+b^2c^2+a^2c^2=49\)

Ta có:

\(a^4+b^4+c^4=\left(a^2+b^2+c^2\right)^2-2a^2b^2-2b^2c^2-2a^2c^2\)\(=14^2-2\left(a^2b^2+b^2c^2+a^2c^2\right)=196-2.49=98\)

Ta có: \(P=\dfrac{ab}{c^2}+\dfrac{bc}{a^2}+\dfrac{ca}{b^2}\)

=> \(\dfrac{ab}{c^2}+\dfrac{bc}{a^2}+\dfrac{ca}{b^2}=abc\left(\dfrac{1+1+1}{a^3+b^3+c^3}\right)\)

Mà ta lại có : \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\Rightarrow\dfrac{1}{a}+\dfrac{1}{b}=-\dfrac{1}{c}\)

\(\dfrac{\Rightarrow1}{a^3}+\dfrac{1}{b^3}=\left(\dfrac{1+1}{a+b}\right)^3-3\dfrac{1}{a}.\dfrac{1}{b}\left(\dfrac{1+1}{a+b}\right)\)

\(\Rightarrow-\dfrac{1^3}{c}+\dfrac{3}{abc}\)

\(\Rightarrow\dfrac{1+1+1}{a^3+b^3+c^3}=\dfrac{3}{abc}\)

\(\dfrac{\Rightarrow bc}{c^2}+\dfrac{ca}{b^2}+\dfrac{ab}{c^2}=3\)

Vậy : \(P=\dfrac{ab}{c^2}+\dfrac{bc}{a^2}+\dfrac{ca}{b^2}\) = 3

a,Sửa lại đề nha bạn:Tính A = \(\dfrac{a^2}{x^2}+\dfrac{b^2}{y^2}+\dfrac{c^2}{z^2}\)

\(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=\dfrac{bcx+acy+abz}{abc}=0\)

\(\Rightarrow bcz+acy+abz=0\)

(2) \(\Rightarrow\dfrac{a^2}{x^2}+\dfrac{b^2}{y^2}+\dfrac{c^2}{z^2}+2\left(\dfrac{ab}{xy}+\dfrac{ac}{xz}+\dfrac{bc}{xz}\right)=4\)\(\Rightarrow A=\dfrac{a^2}{x^2}+\dfrac{b^2}{y^2}+\dfrac{c^2}{z^2}=4-2.\left(\dfrac{abz+acy+bcz}{xyz}\right)=4\)b, \(a+b+c=0\Rightarrow a+b=-c\)

\(\Rightarrow a^2+2ab+b^2=c^2\Rightarrow a^2+b^2-c^2=-2ab\)Tương tự: \(b^2+c^2-a^2=-2bc\)

\(c^2+a^2-b^2=-2bc\)

Vậy \(B=\dfrac{ab}{-2ab}+\dfrac{bc}{-2bc}+\dfrac{ac}{-2ac}=\dfrac{-3}{2}\)

Ta có: \(A=a\left(a^2-bc\right)+b\left(b^2-ac\right)+c\left(c^2-ab\right)=0\)

\(\Rightarrow A=a^3+b^3+c^3-3abc=0\) \(\Rightarrow A=\left(a+b\right)^3+c^3-3ab\left(a+b\right)-3abc=0\)

\(\Rightarrow A=\left(a+b+c\right)\left[\left(a+b\right)^2-c\left(a+b\right)+c^2\right]-3ab\left(a+b+c\right)=0\)

\(\Rightarrow A=\left(a+b+c\right)\left(a^2+b^2+c^2-ab-bc-ac\right)=0\)

Vì \(a+b+c\ne0\Rightarrow a^2+b^2+c^2-ab-ac-bc=0\)

Xét \(M=a^2+b^2+c^2-ab-ac-bc=0\)

\(\Rightarrow2M=2a^2+2b^2+2c^2-2ab-2ac-2bc=0\)

\(\Rightarrow2M=\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)

Vì \(\left(a-b\right)^2\ge0;\left(b-c\right)^2\ge0;\left(c-a\right)^2\ge0\forall a,b,c\)

\(\Rightarrow a-b=0;b-c=0;c-a=0\) \(\Rightarrow a=b=c\)

\(\Rightarrow P=\frac{a^2}{b^2}+\frac{b^2}{c^2}+\frac{c^2}{a^2}=1+1+1=3\) 

Cho mình sửa đề một chút thôi nha mình tin chắc là đề bạn sai rồi

Cho a,b,c thỏa mãn : \(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\) Tính giá trị biểu thức N = \(\dfrac{bc}{a^2}+\dfrac{ac}{b^2}+\dfrac{ab}{c^2}\)

Ta có :

\(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=0\Rightarrow\left(\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}\right)^3=0\)

\(\Rightarrow\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}=\dfrac{3}{abc}\)

Ta lại có :

\(N=\dfrac{bc}{a^2}+\dfrac{ac}{b^2}+\dfrac{ab}{c^2}=\dfrac{abc}{a^3}+\dfrac{abc}{b^3}+\dfrac{abc}{c^3}\)

\(\Leftrightarrow N=abc\left(\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3}\right)=abc\times\dfrac{3}{abc}=3\)

Chúc bạn học tốt =))

bc/a^2 + ac/b^2 + ab/c^2=abc(1/a^3 + 1/b^3 + 1/c^3)
mà 1/a + 1/b + 1/c = 0
=> 1/a + 1/b=-1/c
=> 1/a^3+1/b^3 = (1/a+1/b)^3 - 3.1/a.1/b(1/a+1/b) = -1/c^3 + 3.1/(abc)
=> 1/a^3 + 1/b^3 + 1/c^3=3/(abc)
=> bc/a^2 + ac/b^2 + ab/c^2=3.