SOLUTION: Please explain how you would solve this equation: 150x^2+120x+24

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Question 132769: Please explain how you would solve this equation:
150x^2+120x+24
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You cannot solve this since it is not an equation. But you can factor



150x%5E2%2B120x%2B24 Start with the given expression


6%2825x%5E2%2B20x%2B4%29 Factor out the GCF 6


Now let's focus on the inner expression 25x%5E2%2B20x%2B4




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Looking at 25x%5E2%2B20x%2B4 we can see that the first term is 25x%5E2 and the last term is 4 where the coefficients are 25 and 4 respectively.

Now multiply the first coefficient 25 and the last coefficient 4 to get 100. Now what two numbers multiply to 100 and add to the middle coefficient 20? Let's list all of the factors of 100:



Factors of 100:
1,2,4,5,10,20,25,50

-1,-2,-4,-5,-10,-20,-25,-50 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 100
1*100
2*50
4*25
5*20
10*10
(-1)*(-100)
(-2)*(-50)
(-4)*(-25)
(-5)*(-20)
(-10)*(-10)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20

First NumberSecond NumberSum
11001+100=101
2502+50=52
4254+25=29
5205+20=25
101010+10=20
-1-100-1+(-100)=-101
-2-50-2+(-50)=-52
-4-25-4+(-25)=-29
-5-20-5+(-20)=-25
-10-10-10+(-10)=-20



From this list we can see that 10 and 10 add up to 20 and multiply to 100


Now looking at the expression 25x%5E2%2B20x%2B4, replace 20x with 10x%2B10x (notice 10x%2B10x adds up to 20x. So it is equivalent to 20x)

25x%5E2%2Bhighlight%2810x%2B10x%29%2B4


Now let's factor 25x%5E2%2B10x%2B10x%2B4 by grouping:


%2825x%5E2%2B10x%29%2B%2810x%2B4%29 Group like terms


5x%285x%2B2%29%2B2%285x%2B2%29 Factor out the GCF of 5x out of the first group. Factor out the GCF of 2 out of the second group


%285x%2B2%29%285x%2B2%29 Since we have a common term of 5x%2B2, we can combine like terms

So 25x%5E2%2B10x%2B10x%2B4 factors to %285x%2B2%29%285x%2B2%29


So this also means that 25x%5E2%2B20x%2B4 factors to %285x%2B2%29%285x%2B2%29 (since 25x%5E2%2B20x%2B4 is equivalent to 25x%5E2%2B10x%2B10x%2B4)


note: %285x%2B2%29%285x%2B2%29 is equivalent to %285x%2B2%29%5E2 since the term 5x%2B2 occurs twice. So 25x%5E2%2B20x%2B4 also factors to %285x%2B2%29%5E2



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So our expression goes from 6%2825x%5E2%2B20x%2B4%29 and factors further to 6%285x%2B2%29%5E2


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Answer:

So 150x%5E2%2B120x%2B24 factors to 6%285x%2B2%29%5E2