Which two numbers add up to 14 and multiply to 8?

For now, let's call the numbers A and B .

You gave us two clues about the numbers.
You said:
-- (A + B) = 14
-- (A · B)  =  8 .

There are a couple of different ways to go about this.
Here's my way:

I'm going to use 'substitution' to solve this pair of equations.

-- Take the first equation and solve it for B:

                                            A + B  =  14

Subtract  A  from each side:         B  =  14 - A

-- Take that and write it in place of B in the second equation:

                               A · B  =  8

                               A · (14 - A)  =  8

Eliminate parentheses:     14A - A² = 8

Subtract  8  from each side:     14A - A² - 8 = 0

Multiply each side by  -1 :       -14A + A² + 8 = 0

Re-arrange the terms:             A² - 14A + 8 = 0     

There's your nice familiar quadratic equation to solve.
A quadratic equation always has two solutions. 
The two solutions to this one are the two numbers you're looking for. 

Looking at the left side, it's pretty obvious that there's no easy way
to factor it.  So you'll just have to use the quadratic formula.

When you solve the quadratic equation, you get these two solutions:

            A  =  7 + √41  =  13.4031 ...  (rounded)
and
            A  =  7 - √41  =    0.5969 ...   (rounded)


When you compare these numbers to  Destanyhale's numbers
in the first answer, you'll find that they're completely different.

Don't get scared !  Destanyhale's method of solution was completely
right, and the two numbers over there are perfectly correct, but
there are two 'minor' problems with that  
-- Destanyhale didn't explain how to find the solution.
-- Destanyhale swapped the sum and the product in the question. 
That is, s/he said they add up to 8 and multiply to 14. 

But, after mis-reading the question, the rest of that solution is well done.