SOLUTION: I need help solving trinomials, sorry if I chose the wrong category here is the problem: -y^2+10y-24.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: I need help solving trinomials, sorry if I chose the wrong category here is the problem: -y^2+10y-24.      Log On


   

Question 243323: I need help solving trinomials, sorry if I chose the wrong category
here is the problem: -y^2+10y-24.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming that you want to factor.


-y%5E2%2B10y-24 Start with the given expression.


-%28y%5E2-10y%2B24%29 Factor out the GCF -1.


Now let's try to factor the inner expression y%5E2-10y%2B24


---------------------------------------------------------------


Looking at the expression y%5E2-10y%2B24, we can see that the first coefficient is 1, the second coefficient is -10, and the last term is 24.


Now multiply the first coefficient 1 by the last term 24 to get %281%29%2824%29=24.


Now the question is: what two whole numbers multiply to 24 (the previous product) and add to the second coefficient -10?


To find these two numbers, we need to list all of the factors of 24 (the previous product).


Factors of 24:
1,2,3,4,6,8,12,24
-1,-2,-3,-4,-6,-8,-12,-24


Note: list the negative of each factor. This will allow us to find all possible combinations.


These factors pair up and multiply to 24.
1*24 = 24
2*12 = 24
3*8 = 24
4*6 = 24
(-1)*(-24) = 24
(-2)*(-12) = 24
(-3)*(-8) = 24
(-4)*(-6) = 24

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -10:


First NumberSecond NumberSum
1241+24=25
2122+12=14
383+8=11
464+6=10
-1-24-1+(-24)=-25
-2-12-2+(-12)=-14
-3-8-3+(-8)=-11
-4-6-4+(-6)=-10



From the table, we can see that the two numbers -4 and -6 add to -10 (the middle coefficient).


So the two numbers -4 and -6 both multiply to 24 and add to -10


Now replace the middle term -10y with -4y-6y. Remember, -4 and -6 add to -10. So this shows us that -4y-6y=-10y.


y%5E2%2Bhighlight%28-4y-6y%29%2B24 Replace the second term -10y with -4y-6y.


%28y%5E2-4y%29%2B%28-6y%2B24%29 Group the terms into two pairs.


y%28y-4%29%2B%28-6y%2B24%29 Factor out the GCF y from the first group.


y%28y-4%29-6%28y-4%29 Factor out 6 from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.


%28y-6%29%28y-4%29 Combine like terms. Or factor out the common term y-4


--------------------------------------------------


So -1%28y%5E2-10y%2B24%29 then factors further to -%28y-6%29%28y-4%29


===============================================================

Answer:


So -y%5E2%2B10y-24 completely factors to -%28y-6%29%28y-4%29.


In other words, -y%5E2%2B10y-24=-%28y-6%29%28y-4%29.


Note: you can check the answer by expanding -%28y-6%29%28y-4%29 to get -y%5E2%2B10y-24 or by graphing the original expression and the answer (the two graphs should be identical).