SOLUTION: factor x^2+2x-3

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Question 214534This question is from textbook blitzer college algebra essential
: factor x^2+2x-3 This question is from textbook blitzer college algebra essential

Answer by jim_thompson5910(35256) About Me  (Show Source):
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Looking at the expression x%5E2%2B2x-3, we can see that the first coefficient is 1, the second coefficient is 2, and the last term is -3.


Now multiply the first coefficient 1 by the last term -3 to get %281%29%28-3%29=-3.


Now the question is: what two whole numbers multiply to -3 (the previous product) and add to the second coefficient 2?


To find these two numbers, we need to list all of the factors of -3 (the previous product).


Factors of -3:
1,3
-1,-3


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


These factors pair up and multiply to -3.
1*(-3) = -3
(-1)*(3) = -3

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 2:


First NumberSecond NumberSum
1-31+(-3)=-2
-13-1+3=2



From the table, we can see that the two numbers -1 and 3 add to 2 (the middle coefficient).


So the two numbers -1 and 3 both multiply to -3 and add to 2


Now replace the middle term 2x with -x%2B3x. Remember, -1 and 3 add to 2. So this shows us that -x%2B3x=2x.


x%5E2%2Bhighlight%28-x%2B3x%29-3 Replace the second term 2x with -x%2B3x.


%28x%5E2-x%29%2B%283x-3%29 Group the terms into two pairs.


x%28x-1%29%2B%283x-3%29 Factor out the GCF x from the first group.


x%28x-1%29%2B3%28x-1%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.


%28x%2B3%29%28x-1%29 Combine like terms. Or factor out the common term x-1


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Answer:


So x%5E2%2B2x-3 factors to %28x%2B3%29%28x-1%29.


In other words, x%5E2%2B2x-3=%28x%2B3%29%28x-1%29.


Note: you can check the answer by expanding %28x%2B3%29%28x-1%29 to get x%5E2%2B2x-3 or by graphing the original expression and the answer (the two graphs should be identical).

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