SOLUTION: Find three consecutive integers whose product is 161 larger than the cube of the smallest integer

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Question 1155188: Find three consecutive integers whose product is 161 larger than the cube of the smallest integer
Found 2 solutions by ewatrrr, MathTherapy:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!


Hi  

 Let n represent the smallest integer  

n(n+1)(n+2) = n^3- 161

n(n^2 + 3n + 2)= n^3- 161

n^3 + 3n^2 + 2n = n^3 + 161

3n^2 + 2n - 161 = 0

green%28n=7%29  Integer solution

Integers 7 , 8 , 9

CHECKING our answer***  7*8*9 = 7^3 + 161 = 504

Wish You the Best in your Studies.




 
Solved by pluggable solver: SOLVE quadratic equation with variable


Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 3x%5E2%2B2x%2B-161+=+0) has the following solutons:

  

  x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

  

  For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

  

  First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%282%29%5E2-4%2A3%2A-161=1936.

  

  Discriminant d=1936 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28-2%2B-sqrt%28+1936+%29%29%2F2%5Ca.

  

    x%5B1%5D+=+%28-%282%29%2Bsqrt%28+1936+%29%29%2F2%5C3+=+7

    x%5B2%5D+=+%28-%282%29-sqrt%28+1936+%29%29%2F2%5C3+=+-7.66666666666667

    

    Quadratic expression 3x%5E2%2B2x%2B-161 can be factored:

  3x%5E2%2B2x%2B-161+=+3%28x-7%29%2A%28x--7.66666666666667%29

  Again, the answer is: 7, -7.66666666666667.
Here's your graph:

graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+3%2Ax%5E2%2B2%2Ax%2B-161+%29

Answer by MathTherapy(10552) About Me  (Show Source):
You can put this solution on YOUR website!

Find three consecutive integers whose product is 161 larger than the cube of the smallest integer
Let the first integer be F

Then other 2 are: F + 1, and F + 2

We then get: matrix%281%2C3%2C+F%28F+%2B+1%29%28F+%2B+2%29%2C+%22=%22%2C+F%5E3+%2B+161%29

matrix%281%2C3%2C+F%28F%5E2+%2B+3F+%2B+2%29%2C+%22=%22%2C+F%5E3+%2B+161%29

matrix%281%2C3%2C+3F%5E2+%2B+2F+-+161%2C+%22=%22%2C+0%29

matrix%281%2C3%2C+3F%5E2+%2B+23F+-+21F+-+161%2C+%22=%22%2C+0%29 ----- Substituting 23F - 21F for 2F

F(3F + 23) - 7(3F + 23) = 0 

F - 7 = 0               OR                    3F + 23 = 0

          OR          3F = - 23 (ignore)