SOLUTION: factor 9x^2-11x-58

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Question 502978: factor 9x^2-11x-58
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Looking at the expression 9x%5E2-11x-58, we can see that the first coefficient is 9, the second coefficient is -11, and the last term is -58.


Now multiply the first coefficient 9 by the last term -58 to get %289%29%28-58%29=-522.


Now the question is: what two whole numbers multiply to -522 (the previous product) and add to the second coefficient -11?


To find these two numbers, we need to list all of the factors of -522 (the previous product).


Factors of -522:
1,2,3,6,9,18,29,58,87,174,261,522
-1,-2,-3,-6,-9,-18,-29,-58,-87,-174,-261,-522


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


These factors pair up and multiply to -522.
1*(-522) = -522
2*(-261) = -522
3*(-174) = -522
6*(-87) = -522
9*(-58) = -522
18*(-29) = -522
(-1)*(522) = -522
(-2)*(261) = -522
(-3)*(174) = -522
(-6)*(87) = -522
(-9)*(58) = -522
(-18)*(29) = -522

Now let's add up each pair of factors to see if one pair adds to the middle coefficient -11:


First NumberSecond NumberSum
1-5221+(-522)=-521
2-2612+(-261)=-259
3-1743+(-174)=-171
6-876+(-87)=-81
9-589+(-58)=-49
18-2918+(-29)=-11
-1522-1+522=521
-2261-2+261=259
-3174-3+174=171
-687-6+87=81
-958-9+58=49
-1829-18+29=11



From the table, we can see that the two numbers 18 and -29 add to -11 (the middle coefficient).


So the two numbers 18 and -29 both multiply to -522 and add to -11


Now replace the middle term -11x with 18x-29x. Remember, 18 and -29 add to -11. So this shows us that 18x-29x=-11x.


9x%5E2%2Bhighlight%2818x-29x%29-58 Replace the second term -11x with 18x-29x.


%289x%5E2%2B18x%29%2B%28-29x-58%29 Group the terms into two pairs.


9x%28x%2B2%29%2B%28-29x-58%29 Factor out the GCF 9x from the first group.


9x%28x%2B2%29-29%28x%2B2%29 Factor out 29 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.


%289x-29%29%28x%2B2%29 Combine like terms. Or factor out the common term x%2B2


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Answer:


So 9x%5E2-11x-58 factors to %289x-29%29%28x%2B2%29.


In other words, 9x%5E2-11x-58=%289x-29%29%28x%2B2%29.


Note: you can check the answer by expanding %289x-29%29%28x%2B2%29 to get 9x%5E2-11x-58 or by graphing the original expression and the answer (the two graphs should be identical).