SOLUTION: 4a^2+4ab-3b^2=(2a+3b)()

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Question 808386: 4a^2+4ab-3b^2=(2a+3b)()
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 4a%5E2%2B4ab-3b%5E2, we can see that the first coefficient is 4, the second coefficient is 4, and the last coefficient is -3.


Now multiply the first coefficient 4 by the last coefficient -3 to get %284%29%28-3%29=-12.


Now the question is: what two whole numbers multiply to -12 (the previous product) and add to the second coefficient 4?


To find these two numbers, we need to list all of the factors of -12 (the previous product).


Factors of -12:
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12


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


These factors pair up and multiply to -12.
1*(-12) = -12
2*(-6) = -12
3*(-4) = -12
(-1)*(12) = -12
(-2)*(6) = -12
(-3)*(4) = -12

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 4:


First NumberSecond NumberSum
1-121+(-12)=-11
2-62+(-6)=-4
3-43+(-4)=-1
-112-1+12=11
-26-2+6=4
-34-3+4=1



From the table, we can see that the two numbers -2 and 6 add to 4 (the middle coefficient).


So the two numbers -2 and 6 both multiply to -12 and add to 4


Now replace the middle term 4ab with -2ab%2B6ab. Remember, -2 and 6 add to 4. So this shows us that -2ab%2B6ab=4ab.


4a%5E2%2Bhighlight%28-2ab%2B6ab%29-3b%5E2 Replace the second term 4ab with -2ab%2B6ab.


%284a%5E2-2ab%29%2B%286ab-3b%5E2%29 Group the terms into two pairs.


2a%282a-b%29%2B%286ab-3b%5E2%29 Factor out the GCF 2a from the first group.


2a%282a-b%29%2B3b%282a-b%29 Factor out 3b 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.


%282a%2B3b%29%282a-b%29 Combine like terms. Or factor out the common term 2a-b


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Answer:


So 4a%5E2%2B4ab-3b%5E2 factors to %282a%2B3b%29%282a-b%29.


In other words, 4a%5E2%2B4ab-3b%5E2=%282a%2B3b%29%282a-b%29.


Note: you can check the answer by expanding %282a%2B3b%29%282a-b%29 to get 4a%5E2%2B4ab-3b%5E2 or by graphing the original expression and the answer (the two graphs should be identical).