SOLUTION: How do I factor 12x^2 - 10x - 42 = 0?

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Question 248816: How do I factor 12x^2 - 10x - 42 = 0?
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

12x%5E2-10x-42 Start with the given expression.


2%286x%5E2-5x-21%29 Factor out the GCF 2.


Now let's try to factor the inner expression 6x%5E2-5x-21


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Looking at the expression 6x%5E2-5x-21, we can see that the first coefficient is 6, the second coefficient is -5, and the last term is -21.


Now multiply the first coefficient 6 by the last term -21 to get %286%29%28-21%29=-126.


Now the question is: what two whole numbers multiply to -126 (the previous product) and add to the second coefficient -5?


To find these two numbers, we need to list all of the factors of -126 (the previous product).


Factors of -126:
1,2,3,6,7,9,14,18,21,42,63,126
-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126


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


These factors pair up and multiply to -126.
1*(-126) = -126
2*(-63) = -126
3*(-42) = -126
6*(-21) = -126
7*(-18) = -126
9*(-14) = -126
(-1)*(126) = -126
(-2)*(63) = -126
(-3)*(42) = -126
(-6)*(21) = -126
(-7)*(18) = -126
(-9)*(14) = -126

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -5:


First NumberSecond NumberSum
1-1261+(-126)=-125
2-632+(-63)=-61
3-423+(-42)=-39
6-216+(-21)=-15
7-187+(-18)=-11
9-149+(-14)=-5
-1126-1+126=125
-263-2+63=61
-342-3+42=39
-621-6+21=15
-718-7+18=11
-914-9+14=5



From the table, we can see that the two numbers 9 and -14 add to -5 (the middle coefficient).


So the two numbers 9 and -14 both multiply to -126 and add to -5


Now replace the middle term -5x with 9x-14x. Remember, 9 and -14 add to -5. So this shows us that 9x-14x=-5x.


6x%5E2%2Bhighlight%289x-14x%29-21 Replace the second term -5x with 9x-14x.


%286x%5E2%2B9x%29%2B%28-14x-21%29 Group the terms into two pairs.


3x%282x%2B3%29%2B%28-14x-21%29 Factor out the GCF 3x from the first group.


3x%282x%2B3%29-7%282x%2B3%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.


%283x-7%29%282x%2B3%29 Combine like terms. Or factor out the common term 2x%2B3


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So 2%286x%5E2-5x-21%29 then factors further to 2%283x-7%29%282x%2B3%29


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Answer:


So 12x%5E2-10x-42 completely factors to 2%283x-7%29%282x%2B3%29.


In other words, 12x%5E2-10x-42=2%283x-7%29%282x%2B3%29.


Note: you can check the answer by expanding 2%283x-7%29%282x%2B3%29 to get 12x%5E2-10x-42 or by graphing the original expression and the answer (the two graphs should be identical).