SOLUTION: Solve this question by factoring (factoring is the part that gets me) x^2+4x-3=0

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Question 259293: Solve this question by factoring (factoring is the part that gets me)

x^2+4x-3=0
Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
We want factors of three that add up to 4
3 only has factors 1,3 which happen to add up to 4
But there is a problem. One of the two 1 and 3 is negative but not both.
So we need factors of 3 whose difference is 4
There is a good chance you copied the signs wrong
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B4x-3, we can see that the first coefficient is 1, the second coefficient is 4, 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 4?



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 4:



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




From the table, we can see that there are no pairs of numbers which add to 4. So x%5E2%2B4x-3 cannot be factored.



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Answer:



So x%5E2%2B4%2Ax-3 doesn't factor at all (over the rational numbers).



So x%5E2%2B4%2Ax-3 is prime.


-4 and +3 would work
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2-4x%2B3, we can see that the first coefficient is 1, the second coefficient is -4, and the last term is 3.



Now multiply the first coefficient 1 by the last term 3 to get %281%29%283%29=3.



Now the question is: what two whole numbers multiply to 3 (the previous product) and add to the second coefficient -4?



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 -4:



First NumberSecond NumberSum
131+3=4
-1-3-1+(-3)=-4




From the table, we can see that the two numbers -1 and -3 add to -4 (the middle coefficient).



So the two numbers -1 and -3 both multiply to 3 and add to -4



Now replace the middle term -4x with -x-3x. Remember, -1 and -3 add to -4. So this shows us that -x-3x=-4x.



x%5E2%2Bhighlight%28-x-3x%29%2B3 Replace the second term -4x with -x-3x.



%28x%5E2-x%29%2B%28-3x%2B3%29 Group the terms into two pairs.



x%28x-1%29%2B%28-3x%2B3%29 Factor out the GCF x from the first group.



x%28x-1%29-3%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-3%29%28x-1%29 Combine like terms. Or factor out the common term x-1



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Answer:



So x%5E2-4%2Ax%2B3 factors to %28x-3%29%28x-1%29.



In other words, x%5E2-4%2Ax%2B3=%28x-3%29%28x-1%29.



Note: you can check the answer by expanding %28x-3%29%28x-1%29 to get x%5E2-4%2Ax%2B3 or by graphing the original expression and the answer (the two graphs should be identical).