SOLUTION: How i solve this problem 10x^2-19x+6

Algebra ->  Expressions-with-variables -> SOLUTION: How i solve this problem 10x^2-19x+6      Log On


   

Question 244746: How i solve this problem 10x^2-19x+6
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 this.



Looking at the expression 10x%5E2-19x%2B6, we can see that the first coefficient is 10, the second coefficient is -19, and the last term is 6.


Now multiply the first coefficient 10 by the last term 6 to get %2810%29%286%29=60.


Now the question is: what two whole numbers multiply to 60 (the previous product) and add to the second coefficient -19?


To find these two numbers, we need to list all of the factors of 60 (the previous product).


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


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


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

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -19:


First NumberSecond NumberSum
1601+60=61
2302+30=32
3203+20=23
4154+15=19
5125+12=17
6106+10=16
-1-60-1+(-60)=-61
-2-30-2+(-30)=-32
-3-20-3+(-20)=-23
-4-15-4+(-15)=-19
-5-12-5+(-12)=-17
-6-10-6+(-10)=-16



From the table, we can see that the two numbers -4 and -15 add to -19 (the middle coefficient).


So the two numbers -4 and -15 both multiply to 60 and add to -19


Now replace the middle term -19x with -4x-15x. Remember, -4 and -15 add to -19. So this shows us that -4x-15x=-19x.


10x%5E2%2Bhighlight%28-4x-15x%29%2B6 Replace the second term -19x with -4x-15x.


%2810x%5E2-4x%29%2B%28-15x%2B6%29 Group the terms into two pairs.


2x%285x-2%29%2B%28-15x%2B6%29 Factor out the GCF 2x from the first group.


2x%285x-2%29-3%285x-2%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.


%282x-3%29%285x-2%29 Combine like terms. Or factor out the common term 5x-2


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


Answer:


So 10x%5E2-19x%2B6 factors to %282x-3%29%285x-2%29.


In other words, 10x%5E2-19x%2B6=%282x-3%29%285x-2%29.


Note: you can check the answer by expanding %282x-3%29%285x-2%29 to get 10x%5E2-19x%2B6 or by graphing the original expression and the answer (the two graphs should be identical).