SOLUTION: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. 8b^2 + 10b - 25

Algebra ->  Expressions -> SOLUTION: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive. 8b^2 + 10b - 25       Log On


   

Question 317057: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive.
8b^2 + 10b - 25
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 8b%5E2%2B10b-25, we can see that the first coefficient is 8, the second coefficient is 10, and the last term is -25.


Now multiply the first coefficient 8 by the last term -25 to get %288%29%28-25%29=-200.


Now the question is: what two whole numbers multiply to -200 (the previous product) and add to the second coefficient 10?


To find these two numbers, we need to list all of the factors of -200 (the previous product).


Factors of -200:
1,2,4,5,8,10,20,25,40,50,100,200
-1,-2,-4,-5,-8,-10,-20,-25,-40,-50,-100,-200


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


These factors pair up and multiply to -200.
1*(-200) = -200
2*(-100) = -200
4*(-50) = -200
5*(-40) = -200
8*(-25) = -200
10*(-20) = -200
(-1)*(200) = -200
(-2)*(100) = -200
(-4)*(50) = -200
(-5)*(40) = -200
(-8)*(25) = -200
(-10)*(20) = -200

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 10:


First NumberSecond NumberSum
1-2001+(-200)=-199
2-1002+(-100)=-98
4-504+(-50)=-46
5-405+(-40)=-35
8-258+(-25)=-17
10-2010+(-20)=-10
-1200-1+200=199
-2100-2+100=98
-450-4+50=46
-540-5+40=35
-825-8+25=17
-1020-10+20=10



From the table, we can see that the two numbers -10 and 20 add to 10 (the middle coefficient).


So the two numbers -10 and 20 both multiply to -200 and add to 10


Now replace the middle term 10b with -10b%2B20b. Remember, -10 and 20 add to 10. So this shows us that -10b%2B20b=10b.


8b%5E2%2Bhighlight%28-10b%2B20b%29-25 Replace the second term 10b with -10b%2B20b.


%288b%5E2-10b%29%2B%2820b-25%29 Group the terms into two pairs.


2b%284b-5%29%2B%2820b-25%29 Factor out the GCF 2b from the first group.


2b%284b-5%29%2B5%284b-5%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.


%282b%2B5%29%284b-5%29 Combine like terms. Or factor out the common term 4b-5


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Answer:


So 8b%5E2%2B10b-25 factors to %282b%2B5%29%284b-5%29.


In other words, 8b%5E2%2B10b-25=%282b%2B5%29%284b-5%29.


Note: you can check the answer by expanding %282b%2B5%29%284b-5%29 to get 8b%5E2%2B10b-25 or by graphing the original expression and the answer (the two graphs should be identical).