Someone pays your $8000 for a 2-year structured settlement of $5000 each year for the next year and the year after that.

So each cash flow would be
(each is 1 year)
CF 0 = $8000
CF 1 = -$5000 (negative because you don't get it)
CF 2 = -$5000 (negative as well because you don't get it)

How do you solve for an implicit interest rate without using a finance calculator and solving for an IRR

I think you can use the math tables for ordinary or future annuity.

As a general "go-to" option, a set of tables as mentioned by Rehmanvohra can get you to a good approximation for most annuities, fairly efficiently.

In a situation such as yours--with only two future cash flows--there's an algebraic approach which nails the precise answer, whether the CFs form an annuity or not.

Note that finding the internal return amounts to finding that rate r which sets the NPV of the CF sequence to zero; i.e...

0\ =\ -5,000\left(\frac{1}{1+r}\right)^2\ -\ 5,000\left(\frac{1}{1+r}\right)\ +\ 8,000

Kinduva funny-looking way to set up a standard NPV format, but I did it that way intentionally, because if we let x fill in for [1 / (1+r)], it morphs to

-5,000x^2\ -\ 5,000x\ +\ 8,000\ =\ 0

Now you've got a good ol' fashioned quadratic that you can solve for x using your weapon of choice (quadratic formula; completing the square). Don't forget to then solve for r, after having found x.

Also keep in mind that one of the quadratic's roots may imply r < 0, which you can disregard as being not applicable to the physical situation at hand.

Thanks a lot really appreciate it