[tex] \frac{ \sqrt{6x-2} }{x-2} =2[/tex]
[tex]( \frac{ \sqrt{6x-2}}{x-2} ) ^{2} = (2) ^{2} [/tex]
[tex] \frac{6x-2}{x ^{2} -4x+4} =4[/tex]
6x - 2 = 4(x² - 4x + 4)
6x - 2 = 4x² -16x + 16
4x² - 16x - 6x + 16 + 2 = 0
4x² - 22x + 18 = 0
2 (2x² - 11x + 9) = 0
Factor 2x² - 11x + 9
(2x - 9) (x + 1) = 0
2x - 9 = 0
2x = 9
2x/2 = 9/2
x = 9/2
x - 1 = 0
x = 1
The number is 9/2 or 1.
Check:
x = 9/2
[tex] \frac{ \sqrt{6(9/2)-2} }{(9/2)-2} =2[/tex]
[tex] \frac{ \sqrt{27-2} }{9/2)-(4/2)} =2[/tex]
[tex] \frac{ \sqrt{25} }{(5/2)} =2[/tex]
[tex] \frac{5}{(5/2)} =2[/tex]
(5) (2/5) = 2
2 = 2 (true)
x = 1
[tex] \frac{ \frac{+}{-} \sqrt{(6)(1)-2} }{1-2} =2[/tex]
[tex] \frac{ \frac{+}{-} \sqrt{4} }{-1} =2[/tex]
Choose the negative root
[tex] \frac{-2}{-1} =2[/tex]
2 = 2 (true)
