Question

What is true about the solution of x^2/2x-6 = 9/6x-18

Answer 1

`\text{The domain}\\2x-6\\eq0\ \wedge\ 6x-18\\eq0\to x\\eq3\\\\(x^2)/(2x-6)=(9)/(6x-18)\\\\(x^2)/(2(x-3))=(9)/(6(x-3))\ \ \ |\cdot6\\\\(3x^2)/(x-3)=(9)/(x-3)\iff3x^2=9\ \ \ |:3\\\\x^2=3\to x=\pm\sqrt3`

Answer 2

You can't divide by zero, so

`2x-6 \\eq 0`

`2x \\eq 6`

`x \\eq 3`

`6x-18 \\eq 0`

`6x \\eq 18`

`x \\eq 3`

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

`(x^2)/(2(x - 3)) = (9)/(6(x - 3))`

(x - 3) can be simplified to give

`(x^2)/(2) = (9)/(6)`

Solving for x

`x^2 = (9 * 2)/(6)`

`x^2 = 3`

`x = \pm √(3)`