What adds up to -2 and multiplies to -10?

Answer 1

The answer is √11 - 1, and -1 - √11.

Here is one method of finding it:

Let a + b = -2, and ab = -10.

Now, (a + b)2 = a2 + 2ab + b2, and

(a - b)2 = a2 - 2ab + b2.

Subtracting, we have

(a + b)2 - (a - b)2 = 4ab = - 40, whence

(a - b)2 = (a + b)2 + 40 = 4 + 40 = 44, and

a - b = √44 = 2√11.

Adding

a + b = -2, and

a - b = 2√11, we have,

2a = 2√11 - 2, giving

a = √11 - 1, and

b = -1 - √11.

The sum of these is clearly -2, and their product is -10.