SOLUTION: Factoring Polynomials Finding Solutions 3t^2+11t-20=0

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Question 171359: Factoring Polynomials
Finding Solutions
3t^2+11t-20=0
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
First, we need to factor 3t%5E2%2B11t-20




Looking at 3t%5E2%2B11t-20 we can see that the first term is 3t%5E2 and the last term is -20 where the coefficients are 3 and -20 respectively.

Now multiply the first coefficient 3 and the last coefficient -20 to get -60. Now what two numbers multiply to -60 and add to the middle coefficient 11? Let's list all of the factors of -60:



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 ...List the negative factors as well. This will allow us to find all possible combinations

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

note: remember, the product of a negative and a positive number is a negative number


Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11

First NumberSecond NumberSum
1-601+(-60)=-59
2-302+(-30)=-28
3-203+(-20)=-17
4-154+(-15)=-11
5-125+(-12)=-7
6-106+(-10)=-4
-160-1+60=59
-230-2+30=28
-320-3+20=17
-415-4+15=11
-512-5+12=7
-610-6+10=4



From this list we can see that -4 and 15 add up to 11 and multiply to -60


Now looking at the expression 3t%5E2%2B11t-20, replace 11t with -4t%2B15t (notice -4t%2B15t adds up to 11t. So it is equivalent to 11t)

3t%5E2%2Bhighlight%28-4t%2B15t%29%2B-20


Now let's factor 3t%5E2-4t%2B15t-20 by grouping:


%283t%5E2-4t%29%2B%2815t-20%29 Group like terms


t%283t-4%29%2B5%283t-4%29 Factor out the GCF of t out of the first group. Factor out the GCF of 5 out of the second group


%28t%2B5%29%283t-4%29 Since we have a common term of 3t-4, we can combine like terms

So 3t%5E2-4t%2B15t-20 factors to %28t%2B5%29%283t-4%29


So this also means that 3t%5E2%2B11t-20 factors to %28t%2B5%29%283t-4%29 (since 3t%5E2%2B11t-20 is equivalent to 3t%5E2-4t%2B15t-20)


So 3t%5E2%2B11t-20 factors to %28t%2B5%29%283t-4%29



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So this means that 3t%5E2%2B11t-20=0 is the same as %28t%2B5%29%283t-4%29=0



%28t%2B5%29%283t-4%29=0 Start with the previous equation.


Now set each factor equal to zero:

t%2B5=0 or 3t-4=0


t=-5 or t=4%2F3 Now solve for t in each case


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Answer:


So the solutions are

t=-5 or t=4%2F3