The sum of 3 numbers is 11, the product of the numbers is 24 and the sum of their squares is 53. What are the numbers, are they integers?

Expert Answers

An illustration of the letter 'A' in a speech bubbles

Ok well if we have the sum of these three things, we can find a cubic equation which can be solved.

Suppose our answers are a,b and c.

(a+b+c)^2 = a^2 + b^2 + c^2 + 2(ab+ac+bc)

So (ab+ac+bc) = ((a+b+c)^2 - (a^2+b^2+c^2))/2

Now (x-a)(x-b)(x-c) = x^3 - (a+b+c)x^2 + (ab+ac+bc)x - abc

So our equation is

x^3 - (a+b+c)x^2 + ((a+b+c)^2-(a^2+b^2+c^2))/2 - abc = 0

In the case above we have

x^3 - 11x^2 + (121-53)/2x - 24 = x^3 - 11x^2 + 34x - 24

We can solve this using analytical methods to get roots = 1,4,6

Approved by eNotes Editorial Team
An illustration of the letter 'A' in a speech bubbles

x+y+z=11

xyz=24

x^2+y^2+z^2 = 53

(1,4,6) is the answer.

I just tried a bunch of values.  I am not sure of how to solve a problem like this in general.  I will think about it, and see if there is a method.

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial