equations with fractions help

Hello,

Hoping someone can help me, i keep trying to solve this equation and i never get the right answer

x + 1/2x = 9/5

my method was to get rid of the fractions so the LCM is 5(2x) ?

so multiply each term my the LCM and I end up with 10x^2 -18x + 5 = 0

but that’s not correct because at the back of the book it says x=2 or x = 1/4

which would mean that the original quadratic equation is 4x^2 -9x + 2 = 0

and than that can be factored into (4x -1) (x-2)

can anyone please tell me were I am going wrong

thank you!