a) ĐKXĐ : \(x\ne\pm a\).

Với \(a=-3\) khi đó ta có pt :

\(A=\frac{x-3}{-3-x}-\frac{x+3}{-3+x}=\frac{-3\left(-9+1\right)}{\left(-3\right)^2-x^2}\)

\(\Leftrightarrow\frac{\left(x-3\right)\left(x+3\right)-\left(x+3\right)\left(-3-x\right)}{\left(-3-x\right)\left(-3+x\right)}+\frac{24}{\left(-3-x\right)\left(-3+x\right)}=0\)

\(\Rightarrow x^2-9-\left(-3x-x^2-9-3x\right)+24=0\)

\(\Leftrightarrow2x^2+6x+24=0\)

\(\Leftrightarrow x^2+3x+12=0\) ( vô nghiệm )

Phần b) tương tự.

\(A=\frac{x+a}{a-x}-\frac{x-a}{a+x}=\frac{a\left(3x+1\right)}{a^2-x^2}\)

\(=\frac{x+a}{a-x}+\frac{x-a}{a+x}=\frac{a\left(3+1\right)}{\left(a-x\right)\left(a+x\right)}\)

\(=\frac{\left(x+a\right)^2+\left(x-a\right)\left(a-x\right)}{\left(a-x\right)\left(a+1\right)}=\frac{a\left(3a+1\right)}{\left(a+x\right)\left(a-x\right)}\)

\(\Leftrightarrow\left(x+a\right)^2+\left(x-a\right)\left(a-x\right)=a\left(3a+1\right)\)

\(\Leftrightarrow x^2+2ax+a^2-ax-x^2-a^2+ax=3a^2+a\)

\(\Leftrightarrow2ax=3a^2+a\)

\(\Leftrightarrow x=\frac{3a^2+a}{2a}\left(a\ne0\right)\)

a) Khi x=-3 => \(x=\frac{3\cdot\left(-3\right)^2-3}{2\left(-3\right)}=-13\)

b) a=1

\(\Leftrightarrow x=\frac{3\cdot1^2+1}{2\cdot1}=2\)

tìm tham số a cho phương trình - 4x - 3 = 4x - 7 nhận x = 2 là nghiệm

a) \(ĐKXĐ:x\ne\pm3\)

Với a = -3

\(\Leftrightarrow A=\frac{x-3}{-3-x}-\frac{x+3}{-3+x}=\frac{-3\left[3.\left(-3\right)+1\right]}{\left(-3\right)^2-x^2}\)

\(\Leftrightarrow\frac{3-x}{x+3}-\frac{x+3}{x-3}=\frac{24}{9-x^2}\)

\(\Leftrightarrow\frac{3-x}{x+3}-\frac{x+3}{x-3}+\frac{24}{x^2-9}=0\)

\(\Leftrightarrow\frac{-\left(x-3\right)^2-\left(x+3\right)^2+24}{x^2-9}=0\)

\(\Leftrightarrow-x^2+6x-9-x^2-6x-9+24=0\)

\(\Leftrightarrow-2x^2+6=0\)

\(\Leftrightarrow x^2=3\)

\(\Leftrightarrow x=\pm\sqrt{3}\)(tm)

Vậy với \(a=-3\Leftrightarrow x\in\left\{\sqrt{3};-\sqrt{3}\right\}\)

b) \(ĐKXĐ:x\ne\pm1\)

Với a = 1

\(\Leftrightarrow A=\frac{x+1}{1-x}-\frac{x-1}{1+x}=\frac{3+1}{1-x^2}\)

\(\Leftrightarrow\frac{x+1}{1-x}-\frac{x-1}{1+x}+\frac{4}{x^2-1}=0\)

\(\Leftrightarrow\frac{-\left(x+1\right)^2-\left(x-1\right)^2+4}{x^2-1}=0\)

\(\Leftrightarrow-x^2-2x-1-x^2+2x-1+4=0\)

\(\Leftrightarrow-2x^2+2=0\)

\(\Leftrightarrow x^2=1\)

\(\Leftrightarrow x=\pm1\)(ktm)

Vậy với \(a=1\Leftrightarrow x\in\varnothing\)

c) \(ĐKXĐ:a\ne\pm\frac{1}{2}\)

Thay \(x=\frac{1}{2}\)vào phương trình, ta đươc :

\(A=\frac{\frac{1}{2}+a}{a-\frac{1}{2}}-\frac{\frac{1}{2}-a}{a+\frac{1}{2}}=\frac{a\left(3a+1\right)}{a^2-\frac{1}{4}}\)

\(\Leftrightarrow\frac{a+\frac{1}{2}}{a-\frac{1}{2}}+\frac{a-\frac{1}{2}}{a+\frac{1}{2}}-\frac{3a^2+a}{a^2-\frac{1}{4}}=0\)

\(\Leftrightarrow\frac{\left(a+\frac{1}{2}\right)^2+\left(a-\frac{1}{2}\right)^2-3a^2-a}{a^2-\frac{1}{4}}=0\)

\(\Leftrightarrow a^2+a+\frac{1}{4}+a^2-a+\frac{1}{4}-3a^2-a=0\)

\(\Leftrightarrow-a^2-a+\frac{1}{2}=0\)

\(\Leftrightarrow a^2+a-\frac{1}{2}=0\)

\(\Leftrightarrow\left(a+\frac{1}{2}\right)^2-\frac{3}{4}=0\)

\(\Leftrightarrow\orbr{\begin{cases}a=\frac{\sqrt{3}}{2}-\frac{1}{2}=\frac{\sqrt{3}-1}{2}\\a=-\frac{\sqrt{3}}{2}-\frac{1}{2}=\frac{-\sqrt{3}-1}{2}\end{cases}}\)(TM)

 Vậy với \(x=\frac{1}{2}\Leftrightarrow a\in\left\{\frac{\sqrt{3}-1}{2};\frac{-\sqrt{3}-1}{2}\right\}\) 

a) Với a=4 thì phương trình bằng \(\frac{x+4}{x+2}+\frac{x-2}{x-4}\)= 2  với đkxđ: \(x\ne2,4\)

Giải phương trình: \(\frac{x+4}{x+2}+\frac{x-2}{x-4}\)= 2 => \(1+\frac{2}{x+2}+1+\frac{2}{x-4}=2\)

=> \(\frac{2}{x+2}+\frac{2}{x-4}=0\Rightarrow\frac{1}{x+2}+\frac{1}{x-4}=0\)

=> \(\frac{\left(x-4\right)+\left(x+2\right)}{\left(x+2\right)\cdot\left(x-4\right)}=0\)=> 2x-2=0 => x=1 (thỏa mãn đkxđ)

Vậy x=1

b) Với x=-1 => \(\frac{a-1}{1}+\frac{-3}{-1-a}=2\)(đkxđ: a không bằng -1)

=> \(\left(a-1\right)+\frac{3}{a+1}=2\)

=> \(\frac{a^2-1+3}{a+1}=2\)=> \(a^2+2=2\left(a+1\right)\Rightarrow a^2-2a=0\)

=> \(a\left(a-2\right)=0\)=> a = (0; 2) (thỏa mãn đkxđ)

Vậy để phương trình có nghiệm x=-1 thì a={0; 2}

\(\text{a) Thay a = 4 vào pt ta có:}\)
      \(\frac{x+4}{x+2}+\frac{x-2}{x-4}=2\)
\(\Leftrightarrow\frac{\left(x-4\right)\left(x+4\right)+\left(x-2\right)\left(x+2\right)}{\left(x+2\right)\left(x-4\right)}=2\)
\(\Leftrightarrow\frac{x^2-16+x^2-4}{x^2-4x+2x-8}=2\)
\(\Leftrightarrow\frac{2x^2-20}{x^2-2x-8}=2\)
\(\Leftrightarrow2x^2-20=2.\left(x^2-2x-8\right)\)
\(\Leftrightarrow2x^2-20=2x^2-4x-16\)
\(\Leftrightarrow2x^2-2x^2+4x=-16+20\)
\(\Leftrightarrow4x=4\)
\(\Leftrightarrow x=1\)

\(\text{b) Thay x = -1 vào pt ta có:}\)
     \(\frac{-1+a}{-1+2}+\frac{-1-2}{-1-a}=2\)
\(\Leftrightarrow\frac{a-1}{1}+\frac{-3}{-\left(a+1\right)}=2\)
\(\Leftrightarrow\left(a-1\right)+\frac{3}{a+1}=2\)
\(\Leftrightarrow\frac{\left(a-1\right)\left(a+1\right)+3}{a+1}=2\)
\(\Leftrightarrow\frac{a^2-1+3}{a+1}=2\)
\(\Leftrightarrow a^2+2=2.\left(a+1\right)\)
\(\Leftrightarrow a^2+2=2a+2\)
\(\Leftrightarrow a^2-2a=2-2\)
\(\Leftrightarrow a\left(a-2\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}a=0\\a-2=0\end{cases}\Leftrightarrow\orbr{\begin{cases}a=0\\a=2\end{cases}}}\)
Vậy để pt có nghiệm là x = 1 thì a = {0 ; 2}
 

\(a.Thay:a=4\Leftrightarrow\frac{x+4}{x+2}+\frac{x-2}{x-4}=2\)

                    \(\Leftrightarrow\frac{\left(x+4\right)\left(x-4\right)}{\left(x+2\right)\left(x-4\right)}+\frac{\left(x-2\right)\left(x+2\right)}{\left(x-4\right)\left(x+2\right)}=\frac{2\left(x+2\right)\left(x-4\right)}{\left(x+2\right)\left(x-4\right)}\)

                    \(\Rightarrow\left(x+4\right)\left(x-4\right)+\left(x-2\right)\left(x+2\right)=2\left(x+2\right)\left(x-4\right)\)

                    \(\Leftrightarrow x^2-4x+4x-16+x^2+2x-2x-4=\left(2x+4\right)\left(x-4\right)\)

                    \(\Leftrightarrow2x^2-20=2x^2-8x+4x-16\)

                    \(\Leftrightarrow2x^2-20-2x^2+8x-4x+16=0\)

                    \(\Leftrightarrow4x-4=0\)

                    \(\Leftrightarrow x=1\)

Đk:\(a\ne\pm x\)

Pt \(\Leftrightarrow\dfrac{\left(a+x\right)^2-\left(x-a\right)\left(a-x\right)}{\left(a-x\right)\left(a+x\right)}=\dfrac{a\left(3a+1\right)}{a^2-x^2}\)

\(\Leftrightarrow\dfrac{2\left(a^2+x^2\right)}{a^2-x^2}=\dfrac{a\left(3a+1\right)}{a^2-x^2}\)

\(\Leftrightarrow2a^2+2x^2=3a^2+a\)

\(\Leftrightarrow a^2+a-2x^2=0\) (1)

Thay \(x=\dfrac{1}{2}\) vào (1) ta được:

\(a^2+a-2\left(\dfrac{1}{2}\right)^2=0\)

\(\Leftrightarrow a^2+a-\dfrac{1}{2}=0\)

\(\Leftrightarrow\left[{}\begin{matrix}a=\dfrac{-1+\sqrt{3}}{2}\\a=\dfrac{-1-\sqrt{3}}{2}\end{matrix}\right.\) (tm)

Vậy...

1. a = 3 thì phương trình trở thành:

\(\frac{x+3}{3-x}-\frac{x-3}{3+x}=\frac{-3\left[3.\left(-3\right)+1\right]}{\left(-3\right)^2}-x^2\)

\(\Leftrightarrow\frac{\left(x+3\right)^2+\left(3-x\right)^2}{\left(3-x\right)\left(3+x\right)}=\frac{-3\left[-9+1\right]}{9}-x^2\)

\(\Leftrightarrow\frac{x^2+6x+9+x^2-6x+9}{\left(3-x\right)\left(3+x\right)}=\frac{-3.\left(-8\right)}{9}-x^2\)

\(\Leftrightarrow\frac{2x^2+18}{9-x^2}=\frac{24}{9}-x^2\)

\(\Leftrightarrow\frac{2x^2+18}{9-x^2}+x^2=\frac{24}{9}\)

\(\Leftrightarrow\frac{2x^2+18+9x^2-x^4}{9-x^2}=\frac{24}{9}\)

\(\Leftrightarrow\frac{11x^2+18-x^4}{9-x^2}=\frac{24}{9}\)

\(\Leftrightarrow99x^2+18-9x^4=216-24x^2\)

\(\Leftrightarrow9x^4-123x^2+198=0\)

Đặt \(x^2=t\left(t\ge0\right)\)

Phương trình trở thành \(9t^2-123t+198=0\)

Ta có \(\Delta=123^2-4.9.198=8001,\sqrt{\Delta}=3\sqrt{889}\)

\(\Rightarrow\orbr{\begin{cases}t=\frac{123+3\sqrt{889}}{18}=\frac{41+\sqrt{889}}{6}\\t=\frac{123-3\sqrt{889}}{18}=\frac{41-\sqrt{889}}{6}\end{cases}}\)

Lúc đó \(\orbr{\begin{cases}x^2=\frac{41+\sqrt{889}}{6}\\x^2=\frac{41-\sqrt{889}}{6}\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=\pm\sqrt{\frac{41+\sqrt{889}}{6}}\\x=\pm\sqrt{\frac{41-\sqrt{889}}{6}}\end{cases}}\)

Vậy pt có 4 nghiệm \(S=\left\{\pm\sqrt{\frac{41+\sqrt{889}}{6}};\pm\sqrt{\frac{41-\sqrt{889}}{6}}\right\}\)

Sửa)):

a = -3 mà ghi lôn a = 3.giải tương tự như 3