SOLUTION: Please help solve: 6x^2+7x-3

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Question 593528: Please help solve:
6x^2+7x-3
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
I'm assuming you want to factor this.


Looking at the expression 6x%5E2%2B7x-3, we can see that the first coefficient is 6, the second coefficient is 7, and the last term is -3.


Now multiply the first coefficient 6 by the last term -3 to get %286%29%28-3%29=-18.


Now the question is: what two whole numbers multiply to -18 (the previous product) and add to the second coefficient 7?


To find these two numbers, we need to list all of the factors of -18 (the previous product).


Factors of -18:
1,2,3,6,9,18
-1,-2,-3,-6,-9,-18


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


These factors pair up and multiply to -18.
1*(-18) = -18
2*(-9) = -18
3*(-6) = -18
(-1)*(18) = -18
(-2)*(9) = -18
(-3)*(6) = -18

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 7:


First NumberSecond NumberSum
1-181+(-18)=-17
2-92+(-9)=-7
3-63+(-6)=-3
-118-1+18=17
-29-2+9=7
-36-3+6=3



From the table, we can see that the two numbers -2 and 9 add to 7 (the middle coefficient).


So the two numbers -2 and 9 both multiply to -18 and add to 7


Now replace the middle term 7x with -2x%2B9x. Remember, -2 and 9 add to 7. So this shows us that -2x%2B9x=7x.


6x%5E2%2Bhighlight%28-2x%2B9x%29-3 Replace the second term 7x with -2x%2B9x.


%286x%5E2-2x%29%2B%289x-3%29 Group the terms into two pairs.


2x%283x-1%29%2B%289x-3%29 Factor out the GCF 2x from the first group.


2x%283x-1%29%2B3%283x-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.


%282x%2B3%29%283x-1%29 Combine like terms. Or factor out the common term 3x-1


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Answer:


So 6x%5E2%2B7x-3 factors to %282x%2B3%29%283x-1%29.


In other words, 6x%5E2%2B7x-3=%282x%2B3%29%283x-1%29.


Note: you can check the answer by expanding %282x%2B3%29%283x-1%29 to get 6x%5E2%2B7x-3 or by graphing the original expression and the answer (the two graphs should be identical).

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