simplify the rational expression t^2=2t-24/t^2-36 and state any restrictions on the variable

Answer: t - 4  —————  t - 6 Step-by-step explanation:            t2 + 2t - 24 Simplify   ————————————              t2 - 36    Trying to factor by splitting the middle term    Factoring  t2 + 2t - 24   The first term is,  t2  its coefficient is  1 . The middle term is,  +2t  its coefficient is  2 . The last term, "the constant", is  -24   Multiply the coefficient of the first term by the constant   1 • -24 = -24   Find two factors of  -24  whose sum equals the coefficient of the middle term, which is   2 .      -24    +    1    =    -23        -12    +    2    =    -10        -8    +    3    =    -5        -6    +    4    =    -2        -4    +    6    =    2    That's it Rewrite the polynomial splitting the middle term using the two factors found in step 2 above,  -4  and  6                       t2 - 4t + 6t - 24   Add up the first 2 terms, pulling out like factors :                    t • (t-4)              Add up the last 2 terms, pulling out common factors :                    6 • (t-4) Step-5 : Add up the four terms of step 4 :                    (t+6)  •  (t-4)             Which is the desired factorization Trying to factor as a Difference of Squares : 1.2      Factoring:  t2-36   Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B) Proof :  (A+B) • (A-B) =         A2 - AB + BA - B2 =         A2 - AB + AB - B2 =         A2 - B2 Note :  AB = BA is the commutative property of multiplication. Note :  - AB + AB equals zero and is therefore eliminated from the expression. Check : 36 is the square of 6 Check :  t2  is the square of  t1   Factorization is :       (t + 6)  •  (t - 6)   Canceling Out :    Cancel out  (t + 6)  which appears on both sides of the fraction line.