Given a value for the product/sum can you always come up with a pair of numbers that gives you that product/sum?

Is this pair unique?

To put it another way: given a number, k, can you always come up with a pair of numbers, a and b, such that a+b = ab = k?

Consider the expression (x-a)(x-b) = x^2 – (a+b)x + ab = x^2 – kx + k

This is zero only when x=a or x=b.

So a and b are the solutions to the equation x^2 – kx + k = 0.

We can find a and b by the standard forumala for a quadratic which gives

a, b = {k +- sqrt(k(k-4)) }/2

Now consider that k(k-4) term.

If it is negative we don’t have a solution, not in real numbers anyway.

If it is zero we have a solution with a = b. This happens if k= 0 or k=4 which gives a=b=0 or a=b=2 respectively.

If it is positive we have a unique solution in the real numbers. k(k-4) is positive when k<0 or k> 4.

So we can always find a and b when k<= 0 or k>= 4.

Blimey! Enough with the quadratics already!

]]>[spoiler]Let the numbers be a and b

ab = a + b

Subtract b from both sides:

ab – b = a

Factor out b:

b(a – 1) = a

Divide both sides by a – 1:

b = a / (a – 1)

Basically for whatever number you give me, I subtract 1 from it and divide it into your original number.

Examples:

a = 3

b = 3/2

a = 7

b = 7/6

a = ?

b = ? / (? – 1)

Now looking at the domain of b / (b – 1), all real numbers are possible except b = 1.

Double-check:

1x = x + 1

x = x + 1

0 = 1 (no solution)

**Answer:
**You can stump me with the number 1.

[/spoiler]