rút gọn biểu thức
\(\frac{a^2-bc}{\left(a+b\right)\left(a+c\right)}+\frac{b^2-ca}{\left(b+c\right)\left(b+a\right)}+\frac{c^2-ab}{\left(c+a\right)\left(c+b\right)}\)
1/rút gọn biểu thức:
\(A=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}+\frac{\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
Rút gọn rồi tính giá trị biểu thức :
\(E=\frac{\left(a-x\right)^2}{a\left(b-a\right)\left(c-a\right)}+\frac{\left(b-x\right)^2}{b\left(a-b\right)\left(c-b\right)}+\frac{\left(c-x\right)^2}{c\left(a-c\right)\left(b-c\right)}\)
Biết : \(1-\frac{x^2}{abc}=0\)
Cho a+b+c=0 và a,b,c khác 0.Rút gọn biểu thức
M=\(\frac{2ab}{a^2+\left(b+c\right)\left(b-c\right)}+\frac{2bc}{b^2+\left(c+a\right)\left(c-a\right)}+\frac{2ca}{c^2+\left(a+b\right)\left(a-b\right)}\)
ta có : a+b+c=0=>a+b=-c ; b+c=-a ; a+c=-b
ta có: M= \(\frac{2ab}{a^2+\left(b+c\right)\left(b-c\right)}+\frac{2bc}{b^2+\left(c+a\right)\left(c-a\right)}+\frac{2ca}{c^2+\left(a+b\right)\left(a-b\right)}\)
M=\(\frac{2ab}{a^2-a\left(b-c\right)}+\frac{2bc}{b^2-b\left(c-a\right)}+\frac{2ca}{c^2-c\left(a-b\right)}\)
M=\(\frac{2ab}{a\left(a-b+c\right)}+\frac{2bc}{b\left(b-c+a\right)}+\frac{2ca}{c\left(c-a+b\right)}\)
M=\(\frac{2ab}{-ab+\left(a+c\right)}+\frac{2bc}{-bc+\left(a+b\right)}+\frac{2ac}{-ac+\left(b+c\right)}\)
M=\(\frac{2ab}{-2ab}+\frac{2bc}{-2bc}+\frac{2ca}{-2ca}\)
M=-1-1-1=-3
Vậy với a+b+c=0 thì M=-3
cho a+b+c=0 và a, b, c đều khác 0. Rút gọn biểu thức:
\(\frac{2ab}{a^2+\left(b+c\right)\left(b-c\right)}+\frac{2bc}{b^2+\left(c+a\right)\left(c-a\right)}+\frac{2ca}{c^2+\left(a+b\right)\left(a-b\right)}\)
1/rút gọn biểu thức:
\(A=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}+\frac{\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
A \(=\frac{2}{a-b}+\frac{2}{b-c}+\frac{2}{c-a}+\frac{\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\)
\(=\frac{2\left(b-c\right)\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}+\frac{2\left(a-b\right)\left(c-a\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}+\frac{2\left(a-b\right)\left(b-c\right)}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}+\frac{\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\)
\(=\frac{2\left(b-c\right)\left(c-a\right)+2\left(a-b\right)\left(c-a\right)+2\left(a-b\right)\left(b-c\right)+\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\)
\(=\frac{2ab+2ac+2bc-2a^2-2b^2-2c^2+\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\)
\(=\frac{-\left(a^2-2ab+b^2\right)-\left(b^2-2bc+c^2\right)-\left(c^2-2ac+a^2\right)+\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\)
\(=\frac{-\left(a-b\right)^2-\left(b-c\right)^2-\left(c-a\right)^2+\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\)
\(=\frac{0}{\left(a-b\right)\left(b-a\right)\left(c-a\right)}\) = 0
Bài 1. Cho a+b+c=0. Đặt P=\(\frac{a-b}{b}+\frac{b-c}{a}+\frac{c-a}{b}\); Q=\(\frac{c}{a-b}+\frac{a}{b-c}+\frac{b}{c-a}\).Tính P.Q
b) Rút gọn rồi tính giá trị biểu thức E=\(\frac{\left(a-x\right)^2}{a\left(b-a\right)\left(c-a\right)}+\frac{\left(b-x\right)^2}{b\left(a-b\right)\left(c-b\right)}+\frac{\left(c-x\right)^2}{c\left(a-c\right)\left(b-c\right)}\)biết \(1-\frac{x^2}{abc}=0\)
Rút gọn biểu thức \(A=\frac{1}{\left(a-b\right)\left(a-c\right)}+\frac{1}{\left(b-a\right)\left(b-c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}\)
\(A=\frac{1}{\left(a-b\right)\left(a-c\right)}+\frac{1}{\left(b-a\right)\left(b-c\right)}+\frac{1}{\left(c-a\right)\left(c-b\right)}\)
\(=\frac{c-b}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}+\frac{a-c}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}+\frac{b-a}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}\)
\(=\frac{c-b+b-a+a-c}{\left(a-b\right)\left(b-c\right)\left(c-a\right)}=0\)
Rút gọn biểu thức sau
\(\frac{a^2\left(b-c\right)+b^2\left(c-a\right)+c^2\left(a-b\right)}{a^2\left(b-c\right)-b^2\left(c+a\right)-c^2\left(a-b\right)+2abc}\)
