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\(ĐKXĐ:a,b,c\ne0\)
\(\frac{x-a}{bc}+\frac{x-b}{ca}+\frac{x-c}{ab}=\frac{2}{a}+\frac{2}{b}+\frac{2}{c}\)
\(\Leftrightarrow\frac{xa-a^2}{abc}+\frac{xb-b^2}{abc}+\frac{xc-c^2}{abc}=\frac{2bc}{abc}+\frac{2ac}{abc}+\frac{2ab}{abc}\)
\(\Leftrightarrow\frac{xa-a^2+xb-b^2+xc-c^2}{abc}=\frac{2bc+2ac+2ab}{abc}\)
\(\Leftrightarrow xa-a^2+xb-b^2+xc-c^2=2bc+2ac+2ab\)
\(\Leftrightarrow xa+xb+xc=2bc+2ac+2ab+a^2+b^2+c^2\)
\(\Leftrightarrow x\left(a+b+c\right)=\left(a+b+c\right)^2\)
\(\Leftrightarrow x=a+b+c\)
Vậy x = a + b + c
\(ĐKXĐ:a,b,c\ne0\)
\(\frac{a+b-x}{c}+\frac{b+c-x}{a}+\frac{c+a-x}{b}+\frac{4x}{a+b+c}=1\)
\(\Leftrightarrow\frac{a+b-x}{c}+\frac{b+c-x}{a}+\frac{c+a-x}{b}=1-\frac{4x}{a+b+c}\)
\(\Leftrightarrow1+\frac{a+b-x}{c}+1+\frac{b+c-x}{a}+1+\frac{c+a-x}{b}=4\)
\(-\frac{4x}{a+b+c}\)
\(\Leftrightarrow\frac{a+b+c-x}{c}+\frac{a+b+c-x}{a}+\frac{a+b+c-x}{b}=\)
\(\frac{4\left(a+b+c\right)}{a+b+c}-\frac{4x}{a+b+c}\)
\(\Leftrightarrow\frac{a+b+c-x}{c}+\frac{a+b+c-x}{a}+\frac{a+b+c-x}{b}=\)
\(\frac{4\left(a+b+c-x\right)}{a+b+c}\)
\(\Leftrightarrow\left(a+b+c-x\right)\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{b}-\frac{4}{a+b+c}\right)=0\)
\(\Rightarrow\left(a+b+c-x\right)=0\)hoặc \(\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{b}-\frac{4}{a+b+c}\right)=0\)
+) Nếu \(\Rightarrow\left(a+b+c-x\right)=0\)thì x = a + b + c
+) Nếu \(\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{b}-\frac{4}{a+b+c}\right)=0\)thì x thỏa mãn với mọi số
Quy đồng rồi phân tích nhân tử bình thường đi
\(\left(x-1\right)\left(x-ab-bc-ca\right)\left(a-b\right)\left(b-c\right)\left(c-a\right)=0\)
a. Với a = -3 ta được:
\(\dfrac{x+3}{x-3}-\dfrac{x-3}{x+3}+\dfrac{27-3}{x^2-9}=0\)
\(\Leftrightarrow\dfrac{\left(x+3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}-\dfrac{\left(x-3\right)\left(x-3\right)}{\left(x-3\right)\left(x+3\right)}+\dfrac{24}{\left(x-3\right)\left(x+3\right)}=0\)
\(\Leftrightarrow x^2+6x+9-x^2+6x-9+24=0\)
\(\Leftrightarrow12x+24=0\)
\(\Leftrightarrow x=-2\)
Giải phương trình :
\(\dfrac{x-a}{x+a}-\dfrac{x+a}{x-a}+\dfrac{3a^2+a}{x^2-a^2}=0\)
a) Với a = -3
\(\dfrac{x-3}{x+3}-\dfrac{x+3}{x-3}+\dfrac{27+3}{x^2-3^2}=0\)
ĐKXĐ : \(\left\{{}\begin{matrix}x+3\ne0\\x-3\ne0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x\ne-3\\x\ne3\end{matrix}\right.\)
Ta có : \(\dfrac{x-3}{x+3}-\dfrac{x+3}{x-3}+\dfrac{27+3}{x^2-3^2}\)
\(\Leftrightarrow\) \(\dfrac{\left(x-3\right)\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}-\dfrac{\left(x+3\right)\left(x+3\right)}{\left(x-3\right)\left(x+3\right)}+\dfrac{27+3}{\left(x+3\right)\left(x-3\right)}=0\)
Khử mẫu ta có : \(\left(x-3\right)^2-\left(x+3\right)^2+27+3=0\)
⇔ \(x^2+6x+9-x^2+6x-9+30=0\)
\(\Leftrightarrow12x+30=0\)
\(\Leftrightarrow12x=-30\)
\(\Leftrightarrow x=-\dfrac{5}{2}\)
Tập nghiệm của pt là: \(S=\left\{-\dfrac{5}{2}\right\}\)
b) Với a = 1
\(\dfrac{x-1}{x+1}-\dfrac{x+1}{x-1}+\dfrac{3+3}{x^2-1}=0\)
ĐKXĐ : \(\left\{{}\begin{matrix}x+1\ne0\\x-1\ne0\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}x\ne-1\\x\ne1\end{matrix}\right.\)
Ta có : \(\dfrac{x-1}{x+1}-\dfrac{x+1}{x-1}+\dfrac{3+3}{x^2-1}=0\)
\(\Leftrightarrow\) \(\dfrac{\left(x-1\right)\left(x-1\right)}{\left(x+1\right)\left(x-1\right)}-\dfrac{\left(x+1\right)\left(x+1\right)}{\left(x-1\right)\left(x+1\right)}+\dfrac{3+3}{\left(x+1\right)\left(x-1\right)}=0\)
Khử mẫu ta có : \(\left(x-1\right)^2-\left(x+1\right)^2+6=0\)
\(\Leftrightarrow x^2+x-1-x^2+x+1+6=0\)
\(\Leftrightarrow2x+6=0\)
\(\Leftrightarrow2x=-6\)
\(\Leftrightarrow x=-3\)
Tập nghiệm của pt là : \(S=\left\{-3\right\}\)
\(c,\frac{x-a-b}{c}-1+\frac{x-b-c}{a}-1+\frac{x-a-c}{b}-1=0.\)
\(\frac{x-a-b-c}{c}+\frac{x-a-b-c}{a}+\frac{x-a-b-c}{b}=0\)
\(\left(x-a-b-c\right)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=0\)
=>\(\orbr{\begin{cases}a+b+c=x\\\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0\end{cases}}\)
Vậy.......
