SOLUTION: What is the completely factored form of 12d^2+14d-6

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: What is the completely factored form of 12d^2+14d-6      Log On


   

Question 422089: What is the completely factored form of 12d^2+14d-6
Found 2 solutions by jim_thompson5910, ewatrrr:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

12d%5E2%2B14d-6 Start with the given expression.


2%286d%5E2%2B7d-3%29 Factor out the GCF 2.


Now let's try to factor the inner expression 6d%5E2%2B7d-3


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


Looking at the expression 6d%5E2%2B7d-3, we can see that the first coefficient is 6, the second coefficient is 7, and the last term is -3.


Now multiply the first coefficient 6 by the last term -3 to get %286%29%28-3%29=-18.


Now the question is: what two whole numbers multiply to -18 (the previous product) and add to the second coefficient 7?


To find these two numbers, we need to list all of the factors of -18 (the previous product).


Factors of -18:
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18


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


These factors pair up and multiply to -18.
1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:


First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3



From the table, we can see that the two numbers -2 and 9 add to 7 (the middle coefficient).


So the two numbers -2 and 9 both multiply to -18 and add to 7


Now replace the middle term 7d with -2d%2B9d. Remember, -2 and 9 add to 7. So this shows us that -2d%2B9d=7d.


6d%5E2%2Bhighlight%28-2d%2B9d%29-3 Replace the second term 7d with -2d%2B9d.


%286d%5E2-2d%29%2B%289d-3%29 Group the terms into two pairs.


2d%283d-1%29%2B%289d-3%29 Factor out the GCF 2d from the first group.


2d%283d-1%29%2B3%283d-1%29 Factor out 3 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.


%282d%2B3%29%283d-1%29 Combine like terms. Or factor out the common term 3d-1


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


So 2%286d%5E2%2B7d-3%29 then factors further to 2%282d%2B3%29%283d-1%29


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


Answer:


So 12d%5E2%2B14d-6 completely factors to 2%282d%2B3%29%283d-1%29.


In other words, 12d%5E2%2B14d-6=2%282d%2B3%29%283d-1%29.


Note: you can check the answer by expanding 2%282d%2B3%29%283d-1%29 to get 12d%5E2%2B14d-6 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim

Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi

Factoring

12d^2+14d-6

2(6d^2-7d-3)

2(3d+1)(2d-3)

Note:SUM of the outside terms(-9d) and the inside terms(2d) = -7d