SOLUTION: y=25x^2+55x+24

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Question 280744: y=25x^2+55x+24
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression , we can see that the first coefficient is , the second coefficient is , and the last term is .



Now multiply the first coefficient by the last term to get .



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



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



Factors of :

1,2,3,4,5,6,8,10,12,15,20,24,25,30,40,50,60,75,100,120,150,200,300,600

-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-20,-24,-25,-30,-40,-50,-60,-75,-100,-120,-150,-200,-300,-600



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



These factors pair up and multiply to .

1*600 = 600
2*300 = 600
3*200 = 600
4*150 = 600
5*120 = 600
6*100 = 600
8*75 = 600
10*60 = 600
12*50 = 600
15*40 = 600
20*30 = 600
24*25 = 600
(-1)*(-600) = 600
(-2)*(-300) = 600
(-3)*(-200) = 600
(-4)*(-150) = 600
(-5)*(-120) = 600
(-6)*(-100) = 600
(-8)*(-75) = 600
(-10)*(-60) = 600
(-12)*(-50) = 600
(-15)*(-40) = 600
(-20)*(-30) = 600
(-24)*(-25) = 600


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



First NumberSecond NumberSum
16001+600=601
23002+300=302
32003+200=203
41504+150=154
51205+120=125
61006+100=106
8758+75=83
106010+60=70
125012+50=62
154015+40=55
203020+30=50
242524+25=49
-1-600-1+(-600)=-601
-2-300-2+(-300)=-302
-3-200-3+(-200)=-203
-4-150-4+(-150)=-154
-5-120-5+(-120)=-125
-6-100-6+(-100)=-106
-8-75-8+(-75)=-83
-10-60-10+(-60)=-70
-12-50-12+(-50)=-62
-15-40-15+(-40)=-55
-20-30-20+(-30)=-50
-24-25-24+(-25)=-49




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



So the two numbers and both multiply to and add to



Now replace the middle term with . Remember, and add to . So this shows us that .



Replace the second term with .



Group the terms into two pairs.



Factor out the GCF from the first group.



Factor out 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.



Combine like terms. Or factor out the common term



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



Answer:



So factors to .



In other words, .



Note: you can check the answer by expanding to get or by graphing the original expression and the answer (the two graphs should be identical).


Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation (in our case ) has the following solutons:



For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=625 is greater than zero. That means that there are two solutions: .




Quadratic expression can be factored:

Again, the answer is: -0.6, -1.6. Here's your graph:
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