SOLUTION: factoring a quadratic with leading coefficient x^2-11x+30

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Question 355042: factoring a quadratic with leading coefficient
x^2-11x+30
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression x%5E2-11x%2B30, we can see that the first coefficient is 1, the second coefficient is -11, and the last term is 30.


Now multiply the first coefficient 1 by the last term 30 to get %281%29%2830%29=30.


Now the question is: what two whole numbers multiply to 30 (the previous product) and add to the second coefficient -11?


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 -11:


First NumberSecond NumberSum
1301+30=31
2152+15=17
3103+10=13
565+6=11
-1-30-1+(-30)=-31
-2-15-2+(-15)=-17
-3-10-3+(-10)=-13
-5-6-5+(-6)=-11



From the table, we can see that the two numbers -5 and -6 add to -11 (the middle coefficient).


So the two numbers -5 and -6 both multiply to 30 and add to -11


Now replace the middle term -11x with -5x-6x. Remember, -5 and -6 add to -11. So this shows us that -5x-6x=-11x.


x%5E2%2Bhighlight%28-5x-6x%29%2B30 Replace the second term -11x with -5x-6x.


%28x%5E2-5x%29%2B%28-6x%2B30%29 Group the terms into two pairs.


x%28x-5%29%2B%28-6x%2B30%29 Factor out the GCF x from the first group.


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


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Answer:


So x%5E2-11x%2B30 factors to %28x-6%29%28x-5%29.


In other words, x%5E2-11x%2B30=%28x-6%29%28x-5%29.


Note: you can check the answer by expanding %28x-6%29%28x-5%29 to get x%5E2-11x%2B30 or by graphing the original expression and the answer (the two graphs should be identical).


If you need more help, email me at jim_thompson5910@hotmail.com

Also, feel free to check out my tutoring website

Jim