[tex]\text{The domain}\\2x-6\neq0\ \wedge\ 6x-18\neq0\to x\neq3\\\\\dfrac{x^2}{2x-6}=\dfrac{9}{6x-18}\\\\\dfrac{x^2}{2(x-3)}=\dfrac{9}{6(x-3)}\ \ \ |\cdot6\\\\\dfrac{3x^2}{x-3}=\dfrac{9}{x-3}\iff3x^2=9\ \ \ |:3\\\\x^2=3\to x=\pm\sqrt3[/tex]
You can't divide by zero, so
[tex]2x-6 \neq 0 [/tex]
[tex]2x \neq 6 [/tex]
[tex]x \neq 3 [/tex]
[tex]6x-18 \neq 0 [/tex]
[tex]6x \neq 18 [/tex]
[tex]x \neq 3 [/tex]
2x-6 can be rewritten as 2(x - 3) and 6x-18 can be rewritten as 6(x - 3). Replacing in the original equation gives
[tex]\frac{x^2}{2(x - 3)} = \frac{9}{6(x - 3)} [/tex]
(x - 3) can be simplified to give
[tex]\frac{x^2}{2} = \frac{9}{6} [/tex]
Solving for x
[tex]x^2 = \frac{9 \times 2}{6} [/tex]
[tex]x^2 = 3 [/tex]
[tex]x = \pm \sqrt{3} [/tex]
