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

Jul 30, 2016

:$\text{ } \frac{1}{2 t} \left(t - 2\right) = \frac{1}{2} - \frac{1}{t}$

#### Explanation:

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

$\textcolor{b l u e}{\text{Consider the numerator}}$

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

$\textcolor{b l u e}{\text{Try 1:") ->(3t-1)(t-4) = 3t^2-12t-4t+4 color(red)(larr" Fail}}$

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")

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Consider the denominator}}$

Factor out $2 t \text{ giving } 2 t \left(3 t - 2\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together}}$

$\frac{3 {t}^{2} - 8 {t}_{4}}{6 {t}^{2} - 4 t} \equiv \frac{\left(t - 2\right) \cancel{\left(3 t - 2\right)}}{2 t \cancel{\left(3 t - 2\right)}}$

giving:$\text{ } \frac{1}{2 t} \left(t - 2\right) = \frac{1}{2} - \frac{1}{t}$