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

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Answer 1 · 2016-07-30T16:07:54

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

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

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