SOLUTION: Factor the following quadric equation: x^2+x-30

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Question 873910: Factor the following quadric equation: x^2+x-30
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2%2Bx-30, we can see that the first coefficient is 1, the second coefficient is 1, and the last term is -30.


Now multiply the first coefficient 1 by the last term -30 to get %281%29%28-30%29=-30.


Now the question is: what two whole numbers multiply to -30 (the previous product) and add to the second coefficient 1?


To find these two numbers, we need to list all of the factors of -30 (the previous product).


Factors of -30:
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30


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


These factors pair up and multiply to -30.
1*(-30) = -30
2*(-15) = -30
3*(-10) = -30
5*(-6) = -30
(-1)*(30) = -30
(-2)*(15) = -30
(-3)*(10) = -30
(-5)*(6) = -30

Now let's add up each pair of factors to see if one pair adds to the middle coefficient 1:


First NumberSecond NumberSum
1-301+(-30)=-29
2-152+(-15)=-13
3-103+(-10)=-7
5-65+(-6)=-1
-130-1+30=29
-215-2+15=13
-310-3+10=7
-56-5+6=1



From the table, we can see that the two numbers -5 and 6 add to 1 (the middle coefficient).


So the two numbers -5 and 6 both multiply to -30 and add to 1


Now replace the middle term 1x with -5x%2B6x. Remember, -5 and 6 add to 1. So this shows us that -5x%2B6x=1x.


x%5E2%2Bhighlight%28-5x%2B6x%29-30 Replace the second term 1x with -5x%2B6x.


%28x%5E2-5x%29%2B%286x-30%29 Group the terms into two pairs.


x%28x-5%29%2B%286x-30%29 Factor out the GCF x from the first group.


x%28x-5%29%2B6%28x-5%29 Factor out 6 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%2B6%29%28x-5%29 Combine like terms. Or factor out the common term x-5


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Answer:


So x%5E2%2Bx-30 factors to %28x%2B6%29%28x-5%29.


In other words, x%5E2%2Bx-30=%28x%2B6%29%28x-5%29.


Note: you can check the answer by expanding %28x%2B6%29%28x-5%29 to get x%5E2%2Bx-30 or by graphing the original expression and the answer (the two graphs should be identical).