$\left(1\right)\because x^{2}=3x$，
$\therefore x^{2}-3x=0$，
$\therefore x\left(x-3\right)=0$，
$\therefore x=0$或$x-3=0$，
解得$x_{1}=0$，$x_{2}=3$；
$(2)\because x^{2}+4x-1=0$，
$\therefore x^{2}+4x=1$，
$\therefore x^{2}+4x+4=5$，
$\therefore \left(x+2\right)^{2}=5$，
$\therefore x+2=±\sqrt{5}$，
解得${x}_{1}=-2-\sqrt{5}，{x}_{2}=-2+\sqrt{5}$
$(3)\because 3x^{2}-2=4x$，
$\therefore 3x^{2}-4x-2=0$，
$\therefore a=3$，$b=-4$，$c=-2$，
$\therefore \Delta =\left(-4\right)^{2}-4\times 3\times \left(-2\right)=40 \gt 0$，
$\therefore x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}=\frac{4±2\sqrt{10}}{6}$，
解得${x}_{1}=\frac{2+\sqrt{10}}{3}，{x}_{2}=\frac{2-\sqrt{10}}{3}$；
$(4)\because 2\left(x-3\right)^{2}=x^{2}-9$，
$\therefore 2\left(x-3\right)^{2}-\left(x+3\right)\left(x-3\right)=0$，
$\therefore \left[2\left(x-3\right)-\left(x+3\right)\right]\left(x-3\right)=0$，
$\therefore \left(x-9\right)\left(x-3\right)=0$，
$\therefore x-3=0$或$x-9=0$，
解得$x_{1}=3$，$x_{2}=9$.

$\left(1\right)\because x^{2}=3x$，
$\therefore x^{2}-3x=0$，
$\therefore x\left(x-3\right)=0$，
$\therefore x=0$或$x-3=0$，
解得$x_{1}=0$，$x_{2}=3$；
$(2)\because x^{2}+4x-1=0$，
$\therefore x^{2}+4x=1$，
$\therefore x^{2}+4x+4=5$，
$\therefore \left(x+2\right)^{2}=5$，
$\therefore x+2=±\sqrt{5}$，
解得${x}_{1}=-2-\sqrt{5}，{x}_{2}=-2+\sqrt{5}$
$(3)\because 3x^{2}-2=4x$，
$\therefore 3x^{2}-4x-2=0$，
$\therefore a=3$，$b=-4$，$c=-2$，
$\therefore \Delta =\left(-4\right)^{2}-4\times 3\times \left(-2\right)=40 \gt 0$，
$\therefore x=\frac{-b±\sqrt{{b}^{2}-4ac}}{2a}=\frac{4±2\sqrt{10}}{6}$，
解得${x}_{1}=\frac{2+\sqrt{10}}{3}，{x}_{2}=\frac{2-\sqrt{10}}{3}$；
$(4)\because 2\left(x-3\right)^{2}=x^{2}-9$，
$\therefore 2\left(x-3\right)^{2}-\left(x+3\right)\left(x-3\right)=0$，
$\therefore \left[2\left(x-3\right)-\left(x+3\right)\right]\left(x-3\right)=0$，
$\therefore \left(x-9\right)\left(x-3\right)=0$，
$\therefore x-3=0$或$x-9=0$，
解得$x_{1}=3$，$x_{2}=9$.