Expert: Paul Klarreich - 9/22/2009

Question
Solve for x. x(x+1)^2*(x+2)= 72

Answer
Questioner: Edwin
Country: Norway
Category: Advanced Math
Private: No
Subject: Advance math>algebra>quadratic equation
Question: Solve for x. x(x+1)^2*(x+2)= 72
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I don't know the context for this question, so my answer might not be what you are looking for.

Since  72 = 8*9 = 2 2 2 3 3, you are looking for consecutive integers, one of which gets used twice.

How about  2 * 3 * 3 * 4, which would mean  x = 2.

(Also, I am sure you noticed that  -2 * -3 * -3 * -4  also work.  In this case, the -4 is the x.)

Now your equation can be multiplied out:

x(x^2 + 2x + 1)(x + 2) = 72

x(x^3 + 4x^2 + 5x  + 2) = 72

x^4 + 4x^3 + 5x^2  + 2x = 72

x^4 + 4x^3 + 5x^2  + 2x - 72 = 0

Now you will want to reduce it using the root x=2.  Synthetic division gives:

1    4    5    2   -72  (2)
    2   12   34    72                                 
-------------------------
1    6   17   36

Now look for more factors, of 36 this time, and by Descartes' rule, they must be negative.  As we saw earlier,  x = -4  does it, and I think now you can finish up.

(Hint: I think the other solutions are imaginary.)