\(\Leftrightarrow\frac{6x+5}{12x+9}-\frac{3x-7}{12x-9}=\frac{4x^2+10x-7}{16x^2-9}.\)

\(\Leftrightarrow\frac{\left(6x+5\right)\left(12x-9\right)-\left(3x-7\right)\left(12x+9\right)}{\left(3.4.x\right)^2-\left(3.3\right)^2}=\frac{4x^2+10x-7}{16x^2-9}\)

\(\Leftrightarrow\frac{72x^2+6x-45-\left(36x^2-57x-63\right)}{3^2\left(16x^2-9\right)}=\frac{4x^2+10x-7}{16x^2-9}\)

ĐK: \(16x^2-9\ne0\Leftrightarrow x^2\ne\left(\frac{3}{4}\right)^2\Rightarrow x\ne\pm\frac{3}{4}\)

\(\Leftrightarrow72x^2+6x-45-36x^2+57x+63=36x^2+90x-63\)

\(\Leftrightarrow27x=81\Leftrightarrow x=3\)

\(\frac{12x^2+30x-21}{16x^2-9}-\frac{3x-7}{3-4x}=\frac{6x+5}{4x+3}\)

ĐKXĐ: \(x\ne\pm\frac{3}{4}\)

\(< =>\frac{12x^2+30x-21}{\left(4x-3\right)\left(4x+3\right)}+\frac{3x-7}{4x-3}=\frac{6x+5}{4x+3}\)

\(=>12x^2+30x-21+\left(3x-7\right)\left(4x+3\right)=\left(6x+5\right)\left(4x-3\right)\)

\(< =>12x^2+30x-21+12x^2-19x-21=24x^2+2x-15\)

\(< =>24x^2+11x-42=24x^2+2x-15\)

\(< =>24x^2+11x-42-24x^2-2x+15=0\)

\(< =>9x-27=0\)

\(< =>x=3\left(TM\right)\)

Tập nghiệm phương trình \(S=\left\{3\right\}\)

\(\frac{12x^2+30x-21}{\left(4x-3\right)\left(4x+3\right)}\)-\(\frac{3x-7}{3-4x}\)=\(\frac{6x+5}{4x+3}\)

\(\frac{12x^2+30x-21}{\left(4x-3\right)\left(4x+3\right)}\)+\(\frac{\left(3x-7\right)\left(4x+3\right)}{\left(4x-3\right)\left(4x+3\right)}\)=\(\frac{\left(6x+5\right)\left(4x-3\right)}{\left(4x-3\right)\left(4x+3\right)}\)

12x²+30x-21+12x²-28x+9x-21=24x²+20x-18x-15

12x²+12x²-24x²+30x-28x+9x-20x+18x=21+21-15

-9x =27

x =\(\frac{27}{-9}\)

x =-3

\(b,\frac{x-3}{x-2}=\frac{5}{\left(x-2\right)\left(x+3\right)}\)ĐKXĐ : \(x\ne2;\ne-3\)

\(\Leftrightarrow\frac{x^2-9}{\left(x-2\right)\left(x+3\right)}=\frac{5}{\left(x-2\right)\left(x+3\right)}\)

\(\Leftrightarrow x^2-9=5\)

\(\Leftrightarrow x^2=14\)

\(x=\sqrt{14}\)

.....

a) \(\left(x+3\right)^2-\left(x-3\right)^2=6x\Leftrightarrow\left(x^2+6x+9\right)-\left(x^2-6x+9\right)=6x\)

\(\Leftrightarrow x^2+6x+9-x^2+6x-9=6x\Leftrightarrow12x=6x\)\(\Leftrightarrow12x-6x=0\Leftrightarrow6x=0\Leftrightarrow x=0\)

Vậy phương trình có tập nghiệm S = { 0 }

b)\(-ĐKXĐ:\hept{\begin{cases}x-2\ne0\\\left(x-2\right)\left(x+3\right)\ne0\end{cases}}\Leftrightarrow\hept{\begin{cases}x-2\ne0\\x+3\ne0\end{cases}}\Leftrightarrow\hept{\begin{cases}x\ne2\\x\ne-3\end{cases}}\)

- Ta có :  \(\frac{x-3}{x-2}=\frac{5}{\left(x-2\right)\left(x+3\right)}\Leftrightarrow\frac{x-3}{x-2}-\frac{5}{\left(x-2\right)\left(x+3\right)}=0\)

\(\Leftrightarrow\frac{\left(x-3\right)\left(x+3\right)-5}{\left(x-2\right)\left(x+3\right)}=0\Leftrightarrow\left(x-3\right)\left(x+3\right)=0\)\(\Leftrightarrow\orbr{\begin{cases}x-3=0\\x+3=0\end{cases}}\Leftrightarrow\orbr{\begin{cases}x=3\left(thoaman\right)\\x=-3\left(kothoaman\right)\end{cases}}\)

Vậy phương trình có tập nghiệm S = { 3 }

\(\dfrac{6x+5}{12x+9}+\dfrac{3x-7}{9-12x}=\dfrac{4x^2+10x-7}{16x^2-9}\)

\(\Leftrightarrow\dfrac{6x+5}{3\left(4x+3\right)}-\dfrac{3x-7}{3\left(4x-3\right)}=\dfrac{12x^2+30x-21}{3\left(4x-3\right)\left(4x+3\right)}\)

\(\Leftrightarrow\left(6x+5\right)\left(4x-3\right)-\left(3x-7\right)\left(4x+3\right)=12x^2+30x-21\)

\(\Leftrightarrow24x^2-18x+20x-15-\left(12x^2+9x-28x-21\right)=12x^2+30x-21\)

\(\Leftrightarrow24x^2+2x-15-12x^2+19x+21=12x^2+30x-21\)

=>31x+6=30x-21

=>x=-27

Lời giải:

a) $(x+3)^2-(x-3)^2=6x+18$

$\Leftrightarrow 12x=6x+18\Leftrightarrow 6x=18\Rightarrow x=3$

b) ĐK:$x\neq 2; x\neq 3$

PT $\Rightarrow x+3=\frac{5}{3-x}$

$\Rightarrow (x+3)(3-x)=5$

$\Rightarrow 9-x^2=5$

$\Rightarrow x^2=4\Rightarrow x=\pm 2$. Kết hợp với ĐKXĐ suy ra $x=-2$

c) ĐKXĐ: $x\neq \frac{\pm 3}{4}$

PT $\Leftrightarrow \frac{12x^2+30x-21}{(4x-3)(4x+3)}-\frac{(3x-7)(3x+4)}{(4x-3)(4x+3)}=\frac{(6x+5)(4x-3)}{(4x-3)(4x+3)}$

$\Rightarrow 12x^2+30x-21-(3x-7)(4x+3)=(6x+5)(4x-3)$

$\Leftrightarrow -24x^2+47x+15=0$

$\Rightarrow x=\frac{47\pm \sqrt{3649}}{48}$

d)

ĐK: $x\neq -1; x\neq 2$

PT $\Leftrightarrow \frac{4(x-2)}{(x+1)(x-2)}-\frac{2(x+1)}{(x-2)(x+1)}=\frac{x+3}{(x+1)(x-2)}$

$\Rightarrow 4(x-2)-2(x+1)=x+3$
$\Rightarrow x=13$ (t.m)

\(\frac{12x+1}{6x-2}-\frac{9x-5}{3x+1}=\frac{108x-36x^2-9}{4\left(9x^2-1\right)}\)

 đkxđ \(x\ne\pm\frac{1}{3}\)

\(\Leftrightarrow\frac{12x+1}{2\left(3x-1\right)}-\frac{9x-5}{3x+1}=\frac{108x-36x^2-9}{4\left(3x-1\right)\left(3x+1\right)}\)

\(\Leftrightarrow\frac{\left(24x+2\right)\left(3x+1\right)}{4\left(3x-1\right)\left(3x+1\right)}-\frac{\left(36x-20\right)\left(3x-1\right)\left(3x+1\right)}{4\left(3x-1\right)\left(3x+1\right)}=\frac{-36x^2+10x-9}{4\left(3x-1\right)\left(3x+1\right)}\)

\(\Leftrightarrow72x^2+6x+24x+2-108x^2+60x+36x-20-108x+36x^2+9=0\)

\(\Leftrightarrow18x-9=0\)

\(\Leftrightarrow18x=9\)

\(\Leftrightarrow x=\frac{1}{2}\left(tmđk\right)\)