Question

How do you simplify `(2t^2+7t-4)/(-2t^2-5t+3)`?

Answer 1

`(t+4)/(-1(t+3))`

Explanation:

Factorize first as described below:

`2t^2+7t-4`

Factor by splitting the middle term

Step-1 : Multiply the coefficient of the first term by the constant `2 xx -4 = -8`

Step-2 : Find two factors of `-8` whose sum equals the coefficient of the middle term, which is `7`.

`-1 + 8 = 7`

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, `-1` and `8`

`2t^2-1t+8t-4`

`t(2t-1)+4(2t-1)`

`(2t-1)(t+4)` -----> Factors!

Now lets do the same for `-2t^2-5t+3`

** Re-write the equation as `color(red)(-1(2t^2+5t-3))`**

Factor by splitting the middle term

Step-1 : Multiply the coefficient of the first term by the constant `2 xx -3 = -6`

Step-2 : Find two factors of `-6` whose sum equals the coefficient of the middle term, which is `5` .

`-1 + 6 = 5`

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step `2` above, `-1` and `6`.

`2t^2-1t+6t-3`

`t(2t-1)+3(2t-1)`

`-1(2t-1)(t+3)` ----- Factors!

So now we get:

`(2t^2+7t-4)/(-2t^2-5t+3)` = `((2t-1)(t+4))/(-1(2t-2)(t+3)`

`((2t-1)(t+4))/(-1(2t-1)(t+3)` = `(cancel(2t-1)(t+4))/(-1cancel(2t-1)(t+3)`

`(t+4)/(-1(t+3))`