SOLUTION: 4b^2-7b-15

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Question 681947: 4b^2-7b-15
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 4b%5E2-7b-15, we can see that the first coefficient is 4, the second coefficient is -7, and the last term is -15.


Now multiply the first coefficient 4 by the last term -15 to get %284%29%28-15%29=-60.


Now the question is: what two whole numbers multiply to -60 (the previous product) and add to the second coefficient -7?


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 -7:


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 the table, we can see that the two numbers 5 and -12 add to -7 (the middle coefficient).


So the two numbers 5 and -12 both multiply to -60 and add to -7


Now replace the middle term -7b with 5b-12b. Remember, 5 and -12 add to -7. So this shows us that 5b-12b=-7b.


4b%5E2%2Bhighlight%285b-12b%29-15 Replace the second term -7b with 5b-12b.


%284b%5E2%2B5b%29%2B%28-12b-15%29 Group the terms into two pairs.


b%284b%2B5%29%2B%28-12b-15%29 Factor out the GCF b from the first group.


b%284b%2B5%29-3%284b%2B5%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.


%28b-3%29%284b%2B5%29 Combine like terms. Or factor out the common term 4b%2B5


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Answer:


So 4b%5E2-7b-15 factors to %28b-3%29%284b%2B5%29.


In other words, 4b%5E2-7b-15=%28b-3%29%284b%2B5%29.


Note: you can check the answer by expanding %28b-3%29%284b%2B5%29 to get 4b%5E2-7b-15 or by graphing the original expression and the answer (the two graphs should be identical).