SOLUTION: Can you please help me factor this equation? 16x^2-86x+30 I removed 2 so I have 2(8x^2-43x+15) I don't know where to go from here...thank you!

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Can you please help me factor this equation? 16x^2-86x+30 I removed 2 so I have 2(8x^2-43x+15) I don't know where to go from here...thank you!      Log On


   

Question 704617: Can you please help me factor this equation?
16x^2-86x+30
I removed 2 so I have 2(8x^2-43x+15)
I don't know where to go from here...thank you!
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

16x%5E2-86x%2B30 Start with the given expression.


2%288x%5E2-43x%2B15%29 Factor out the GCF 2.


Now let's try to factor the inner expression 8x%5E2-43x%2B15


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


Looking at the expression 8x%5E2-43x%2B15, we can see that the first coefficient is 8, the second coefficient is -43, and the last term is 15.


Now multiply the first coefficient 8 by the last term 15 to get %288%29%2815%29=120.


Now the question is: what two whole numbers multiply to 120 (the previous product) and add to the second coefficient -43?


To find these two numbers, we need to list all of the factors of 120 (the previous product).


Factors of 120:
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120
-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-30,-40,-60,-120


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


These factors pair up and multiply to 120.
1*120 = 120
2*60 = 120
3*40 = 120
4*30 = 120
5*24 = 120
6*20 = 120
8*15 = 120
10*12 = 120
(-1)*(-120) = 120
(-2)*(-60) = 120
(-3)*(-40) = 120
(-4)*(-30) = 120
(-5)*(-24) = 120
(-6)*(-20) = 120
(-8)*(-15) = 120
(-10)*(-12) = 120

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -43:


First NumberSecond NumberSum
11201+120=121
2602+60=62
3403+40=43
4304+30=34
5245+24=29
6206+20=26
8158+15=23
101210+12=22
-1-120-1+(-120)=-121
-2-60-2+(-60)=-62
-3-40-3+(-40)=-43
-4-30-4+(-30)=-34
-5-24-5+(-24)=-29
-6-20-6+(-20)=-26
-8-15-8+(-15)=-23
-10-12-10+(-12)=-22



From the table, we can see that the two numbers -3 and -40 add to -43 (the middle coefficient).


So the two numbers -3 and -40 both multiply to 120 and add to -43


Now replace the middle term -43x with -3x-40x. Remember, -3 and -40 add to -43. So this shows us that -3x-40x=-43x.


8x%5E2%2Bhighlight%28-3x-40x%29%2B15 Replace the second term -43x with -3x-40x.


%288x%5E2-3x%29%2B%28-40x%2B15%29 Group the terms into two pairs.


x%288x-3%29%2B%28-40x%2B15%29 Factor out the GCF x from the first group.


x%288x-3%29-5%288x-3%29 Factor out 5 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.


%28x-5%29%288x-3%29 Combine like terms. Or factor out the common term 8x-3


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


So 2%288x%5E2-43x%2B15%29 then factors further to 2%28x-5%29%288x-3%29


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


Answer:


So 16x%5E2-86x%2B30 completely factors to 2%28x-5%29%288x-3%29.


In other words, 16x%5E2-86x%2B30=2%28x-5%29%288x-3%29.


Note: you can check the answer by expanding 2%28x-5%29%288x-3%29 to get 16x%5E2-86x%2B30 or by graphing the original expression and the answer (the two graphs should be identical).

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

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

To find other ways to contact me, you can also visit my website: http://www.freewebs.com/jimthompson5910/home.html

Thanks,

Jim

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