Question 282422: Solve using the five-step problem-solving process. Show all steps necessary to arrive at your solution.
The product of two consecutive positive integers is 272. Find the integers.
Found 2 solutions by richwmiller, solver91311:
Answer by richwmiller(9135)   (Show Source):
You can put this solution on YOUR website!x*(x+1)=272
x^2+x-272=0
16 and 17
| Solved by pluggable solver: Factoring using the AC method (Factor by Grouping) |
In order to factor  , first multiply the leading coefficient 1 and the last term -272 to get -272. Now we need to ask ourselves: What two numbers multiply to -272 and add to 1? Lets find out by listing all of the possible factors of -272
Factors:
1,2,4,8,16,17,34,68,136,272,
-1,-2,-4,-8,-16,-17,-34,-68,-136,-272, List the negative factors as well. This will allow us to find all possible combinations
These factors pair up to multiply to -272.
(-1)*(272)=-272
(-2)*(136)=-272
(-4)*(68)=-272
(-8)*(34)=-272
(-16)*(17)=-272
Now which of these pairs add to 1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 1
| First Number | | | Second Number | | | Sum | | 1 | | | -272 | || | 1+(-272)=-271 | | 2 | | | -136 | || | 2+(-136)=-134 | | 4 | | | -68 | || | 4+(-68)=-64 | | 8 | | | -34 | || | 8+(-34)=-26 | | 16 | | | -17 | || | 16+(-17)=-1 | | -1 | | | 272 | || | (-1)+272=271 | | -2 | | | 136 | || | (-2)+136=134 | | -4 | | | 68 | || | (-4)+68=64 | | -8 | | | 34 | || | (-8)+34=26 | | -16 | | | 17 | || | (-16)+17=1 |
We can see from the table that -16 and 17 add to 1. So the two numbers that multiply to -272 and add to 1 are: -16 and 17
So the original quadratic
breaks down to this (just replace  with the two numbers that multiply to -272 and add to 1, which are: -16 and 17)
 Replace with 
Group the first two terms together and the last two terms together like this:
Factor a 1x out of the first group and factor a 17 out of the second group.
Now since we have a common term  we can combine the two terms.
 Combine like terms.
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Answer:
So the quadratic factors to
Notice how foils back to our original problem . This verifies our answer. | |
| Solved by pluggable solver: SOLVE quadratic equation with variable |
Quadratic equation  (in our case  ) has the following solutons:
For these solutions to exist, the discriminant  should not be a negative number.
First, we need to compute the discriminant :  .
Discriminant d=1089 is greater than zero. That means that there are two solutions:  .
Quadratic expression  can be factored:
Again, the answer is: 16, -17.
Here's your graph:
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Answer by solver91311(16872)  (Show Source):
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