Question

simplify (t²+6t)/(t-1) - 7/(t-1)

and explain please !

Answer 1

First, as both fractions have a common denominator, we can simply combine them to form \[\frac{ t^2+6t-7 }{ t-1 }\] Then, we can factor the quadratic in the numerator using the AC method (or whichever method you prefer), thus making: \[\frac{ (t+7)(t-1) }{ t-1 }\] Finally, we can cancel terms, leaving a final answer of \[t+7\]

Answer 2

ugh i dont get the AC method lol, can you teach me ?

Answer 3

@ssilvestro

Answer 4

I'm sure you'll find much more thorough explanations on the Internet than what I can type here, but this is the gist of it: a general quadratic takes the form\[ax^2+bx+c\]First, you multiply the a and c terms. List all of the pairs of factors for this number. Second, try to find a pair of factors such that they add to equal the b coefficient. An example would be better.
Let's try to factor:\[2x^2-5x-3\]Multiplying the first and last coefficients, we get a product of -6 (2 * -3 = -6). Next, we list all of the factors of 6 (disregard the fact that it's negative at this point). The only factor pairs are (1,6) and (2,3). Can we find such a combination that when added or subtracted together equal the middle coefficient of our quadratic (that is, -5)? We can: (1,6) does the job, because 1-6=-5, and 1 * -6 = -6. This satisfies both of our requirements. Now, we can rewrite our quadratic:
\[2x^2-5x-3=2x^2+x-6x-3\]Do you see how these are equivalent? Now we can factor by grouping:
\[2x^2+x-6x-3=x(2x+1)-3(2x+1)=(2x+1)(x-3)\]This is our final, fully factored form. You may be wondering though why we didn't choose the factor pair (2,3). This pair doesn't work because, although -2 - 3 = -5, -2 * -3 = 6, and we needed our 6 to be negative.
In other words, you want your factor pair to satisfy two rules:
Their sum must equal the middle coefficient (that is, b), and their product must equal a*c. You can choose whatever signage you want for them, as long as they satisfy those two rules. For instance, I chose to make the 6 negative in my choice of factor pair above. This is OK as, again, 1-6=-5=b and 1 * -6 = -6 = a*c.
I'm not sure how clear this explanation is, so my advice would be to check out some YouTube videos or, preferably, refer to your textbook, as I'm sure it illustrates the process more clearly.