SOLUTION: 4x^2-49=0

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Question 260240: 4x^2-49=0
Found 2 solutions by stanbon, richwmiller:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
4x^2-49=0
Factor:
(2x-7)(2x+7) = 0
x = 7/2 or x = -7/2
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Cheers,
Stan H.

Answer by richwmiller(17219) About Me  (Show Source):
You can put this solution on YOUR website!
find factors
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression 4x%5E2%2B0x-49, we can see that the first coefficient is 4, the second coefficient is 0, and the last term is -49.



Now multiply the first coefficient 4 by the last term -49 to get %284%29%28-49%29=-196.



Now the question is: what two whole numbers multiply to -196 (the previous product) and add to the second coefficient 0?



To find these two numbers, we need to list all of the factors of -196 (the previous product).



Factors of -196:

1,2,4,7,14,28,49,98,196

-1,-2,-4,-7,-14,-28,-49,-98,-196



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



These factors pair up and multiply to -196.

1*(-196) = -196
2*(-98) = -196
4*(-49) = -196
7*(-28) = -196
14*(-14) = -196
(-1)*(196) = -196
(-2)*(98) = -196
(-4)*(49) = -196
(-7)*(28) = -196
(-14)*(14) = -196


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 0:



First NumberSecond NumberSum
1-1961+(-196)=-195
2-982+(-98)=-96
4-494+(-49)=-45
7-287+(-28)=-21
14-1414+(-14)=0
-1196-1+196=195
-298-2+98=96
-449-4+49=45
-728-7+28=21
-1414-14+14=0




From the table, we can see that the two numbers -14 and 14 add to 0 (the middle coefficient).



So the two numbers -14 and 14 both multiply to -196 and add to 0



Now replace the middle term 0x with -14x%2B14x. Remember, -14 and 14 add to 0. So this shows us that -14x%2B14x=0x.



4x%5E2%2Bhighlight%28-14x%2B14x%29-49 Replace the second term 0x with -14x%2B14x.



%284x%5E2-14x%29%2B%2814x-49%29 Group the terms into two pairs.



2x%282x-7%29%2B%2814x-49%29 Factor out the GCF 2x from the first group.



2x%282x-7%29%2B7%282x-7%29 Factor out 7 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.



%282x%2B7%29%282x-7%29 Combine like terms. Or factor out the common term 2x-7



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Answer:



So 4%2Ax%5E2%2B0%2Ax-49 factors to %282x%2B7%29%282x-7%29.



In other words, 4%2Ax%5E2%2B0%2Ax-49=%282x%2B7%29%282x-7%29.



Note: you can check the answer by expanding %282x%2B7%29%282x-7%29 to get 4%2Ax%5E2%2B0%2Ax-49 or by graphing the original expression and the answer (the two graphs should be identical).