SOLUTION: Factor 2x^2+5x-25

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Question 91478: Factor

2x^2+5x-25
Answer by jim_thompson5910(35256) About Me  (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 2x%5E2%2B5x-25, we can see that the first coefficient is 2, the second coefficient is 5, and the last term is -25.



Now multiply the first coefficient 2 by the last term -25 to get %282%29%28-25%29=-50.



Now the question is: what two whole numbers multiply to -50 (the previous product) and add to the second coefficient 5?



To find these two numbers, we need to list all of the factors of -50 (the previous product).



Factors of -50:

1,2,5,10,25,50

-1,-2,-5,-10,-25,-50



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



These factors pair up and multiply to -50.

1*(-50) = -50
2*(-25) = -50
5*(-10) = -50
(-1)*(50) = -50
(-2)*(25) = -50
(-5)*(10) = -50


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 5:



First NumberSecond NumberSum
1-501+(-50)=-49
2-252+(-25)=-23
5-105+(-10)=-5
-150-1+50=49
-225-2+25=23
-510-5+10=5




From the table, we can see that the two numbers -5 and 10 add to 5 (the middle coefficient).



So the two numbers -5 and 10 both multiply to -50 and add to 5



Now replace the middle term 5x with -5x%2B10x. Remember, -5 and 10 add to 5. So this shows us that -5x%2B10x=5x.



2x%5E2%2Bhighlight%28-5x%2B10x%29-25 Replace the second term 5x with -5x%2B10x.



%282x%5E2-5x%29%2B%2810x-25%29 Group the terms into two pairs.



x%282x-5%29%2B%2810x-25%29 Factor out the GCF x from the first group.



x%282x-5%29%2B5%282x-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.



%28x%2B5%29%282x-5%29 Combine like terms. Or factor out the common term 2x-5



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Answer:



So 2%2Ax%5E2%2B5%2Ax-25 factors to %28x%2B5%29%282x-5%29.



In other words, 2%2Ax%5E2%2B5%2Ax-25=%28x%2B5%29%282x-5%29.



Note: you can check the answer by expanding %28x%2B5%29%282x-5%29 to get 2%2Ax%5E2%2B5%2Ax-25 or by graphing the original expression and the answer (the two graphs should be identical).