$a)x-7\ne 0\iff x\ne 7$

$D=\mathbb{R}-\left\{7\right\}$

$\frac{x+7}{x-7}=0\mid \cdot \left(x-7\right)$

$x+7=0\mid -7$

$x=-7\wedge x\in D$

$x=-7$

$b)x^2-5\ne 0\iff x^2\ne 5\iff \left(x\ne \sqrt{5}\wedge x\ne -\sqrt{5}\right)$

$D=\mathbb{R}-\left\{-\sqrt{5};\sqrt{5}\right\}$

$\frac{x^2}{x^2-5}=0\mid \cdot \left(x^2-5\right)$

$x^2=0$

$x=0\wedge x\in D$

$x=0$

$c)x\ne 0$

$D=\mathbb{R}-\left\{0\right\}$

$\frac{x^2+2x}{x}=0\mid \cdot x$

$x^2+2x=0$

$x\left(x+2\right)=0$

$\left(x=0\vee x+2=0\right)\wedge x\in D$

$\left(x=0\vee x=-2\right)\wedge x\in \mathbb{R}-\left\{0\right\}$

$x=-2$

$d)x^2-x\ne 0$

$x\left(x-1\right)\ne 0$

$x\ne 0\wedge x-1\ne 0$

$x\ne 0\wedge x\ne 1$

$D=\mathbb{R}-\left\{0;1\right\}$

$\frac{x^3-4x^2+3x}{x^2-x}=0\mid \cdot \left(x^2-x\right)$

$x^3-4x^2+3x=0$

$x\left(x^2-4x+3\right)=0$

$\Delta ={\left(-4\right)}^2-4\cdot 1\cdot 3=16-12=4$

$\sqrt{\Delta }=\sqrt{4}=2$

$x_1=\frac{4-2}{2}=1\vee x_2=\frac{4+2}{2}=3$

$\left(x=0\vee x=1\vee x=3\right)\wedge x\in D$

$\left(x=0\vee x=1\vee x=3\right)\wedge x\in \mathbb{R}-\left\{0;1\right\}$

$x=3$