Question

How do you simplify the expression `(3t^2-8t+4)/(6t^2-4t)`?

Answer 1

:`" "1/(2t)(t-2) = 1/2-1/t`

Explanation:

The first thing to try is to see if there are any common factors you can cancel out.

`color(blue)("Consider the numerator")`

3 is prime so I can not factor out any constants from all of the numerator

`color(blue)("Try 1:") ->(3t-1)(t-4) = 3t^2-12t-4t+4 color(red)(larr" Fail")`

`color(blue)("Try 2:") " Write as: "color(green)( 3t^2-6t-2t+4)color(purple)( -> 3t(t-2)-2(t-2))`
Giving: `" "(t-2)(3t-2)color(red)(" "larr" Works")`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Consider the denominator")`

Factor out `2t" giving "2t(3t-2)`

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Putting it all together")`

`(3t^2-8t_4)/(6t^2-4t) -= ((t-2)cancel((3t-2)))/(2tcancel((3t-2)))`

giving:`" "1/(2t)(t-2) = 1/2-1/t`