SOLUTION: Find three consecutive odd integers such that the product of the second and the third one is 99.

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Question 440143: Find three consecutive odd integers such that the product of the second and the third one is 99.
Answer by rwm(914) About Me  (Show Source):
You can put this solution on YOUR website!
x,x+2 and x+4
(x+2)*(x+4)=99
x^2+6x-91 = 0
(x+13)*(x-7)=0
x = -13
x = +7
Solved by pluggable solver: Factoring using the AC method (Factor by Grouping)


Looking at the expression x%5E2%2B6x-91, we can see that the first coefficient is 1, the second coefficient is 6, and the last term is -91.



Now multiply the first coefficient 1 by the last term -91 to get %281%29%28-91%29=-91.



Now the question is: what two whole numbers multiply to -91 (the previous product) and add to the second coefficient 6?



To find these two numbers, we need to list all of the factors of -91 (the previous product).



Factors of -91:

1,7,13,91

-1,-7,-13,-91



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



These factors pair up and multiply to -91.

1*(-91) = -91
7*(-13) = -91
(-1)*(91) = -91
(-7)*(13) = -91


Now let's add up each pair of factors to see if one pair adds to the middle coefficient 6:



First NumberSecond NumberSum
1-911+(-91)=-90
7-137+(-13)=-6
-191-1+91=90
-713-7+13=6




From the table, we can see that the two numbers -7 and 13 add to 6 (the middle coefficient).



So the two numbers -7 and 13 both multiply to -91 and add to 6



Now replace the middle term 6x with -7x%2B13x. Remember, -7 and 13 add to 6. So this shows us that -7x%2B13x=6x.



x%5E2%2Bhighlight%28-7x%2B13x%29-91 Replace the second term 6x with -7x%2B13x.



%28x%5E2-7x%29%2B%2813x-91%29 Group the terms into two pairs.



x%28x-7%29%2B%2813x-91%29 Factor out the GCF x from the first group.



x%28x-7%29%2B13%28x-7%29 Factor out 13 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%2B13%29%28x-7%29 Combine like terms. Or factor out the common term x-7



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Answer:



So x%5E2%2B6%2Ax-91 factors to %28x%2B13%29%28x-7%29.



In other words, x%5E2%2B6%2Ax-91=%28x%2B13%29%28x-7%29.



Note: you can check the answer by expanding %28x%2B13%29%28x-7%29 to get x%5E2%2B6%2Ax-91 or by graphing the original expression and the answer (the two graphs should be identical).