SOLUTION: The product of three integers is 90. The second number is twice the first number. The third number is two more than the first number. What are the three numbers? Use a polynomial a

Question 1177626: The product of three integers is 90. The second number is twice the first number. The third number is two more than the first number. What are the three numbers? Use a polynomial and the calculator to answer the question.
Found 4 solutions by mananth, josgarithmetic, greenestamps, ikleyn:
Answer by mananth(16946)   (Show Source): You can put this solution on YOUR website!
The product of three integers is 90.
let the numbers be x,y,z
The second number is twice the first number.
y = 2x
The third number is two more than the first number.
z = x+2
Substitute for y and for z and solve for x
x(2x)(x+2) = 90
2x^3+4x^2-90 = 0
---
x^3 + 2x^2 - 45 = 0-
graph the cubic and you will get a solution x = 3
y = 2x = 6
z = x+2 = 5
Answer by josgarithmetic(39620)   (Show Source): You can put this solution on YOUR website!

FIRST n SECOND 2n THIRD n+2 PRODUCT 90

-

Try looking for factorizations.

The numbers are 3, 5, 6.
Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!

Formal algebra doesn't really help to solve this problem.

One response you have received says form the equation and look for solutions informally.

The other response says graph the equation to find the solution.

Both solution methods are valid; but the formal algebra parts of the solutions didn't get you any closer to the answer -- you still had to "look for" the answer, or get it by using a graphing utility.

This problem is best solved informally from the beginning, without trying to use formal algebra.

The prime factorization of 90 is 2*3*3*5. Combine two of those factors in a way that gives you three integers that satisfy the conditions of the problem. The way to do that is easily found: (2*3)*3*5 = 6*3*5. 6 is twice 3; and 5 is 2 more than 3.

ANSWER: 3, 6, and 5

Answer by ikleyn(52812)   (Show Source): You can put this solution on YOUR website!
.

When you guess the solution, then, even if you guessed it correctly, the question remains if the guessed solution is unique.

Without answering this remaining question, the solution can not be considered as perfectly correct/completed.

But in this problem, and in many other similar problems, there is a brilliant and cheep way to prove the uniqueness.

Notice that the function   f(x) = x*(2x)*(x+2)   is monotonic in positive domain,
since it is the product of three monotonic functions.

Therefore, if you guessed x= 3 as the solution for   x*(2x)*(x+2) = 90,   you can be sure that this solution is unique

(at least, in the positive domain x > 0 of real numbers).

So, the guessed solution x = 3 is a UNIQUE solution in the positive domain.

To learn this TRUTH is just a good compensation for reading this post.

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