# 1981 AHSME Problems/Problem 29

## Problem

If , then the sum of the real solutions of

is equal to

## Solution

A solution is available here. Pull up find, and put in "Since x is the principal", and you will arrive at the solution.

It's not super clear, and there's some black stuff over it, but its legible.

The solution in the above file/pdf is the following. I tried my best to match it verbatim, but I had to guess at some things. I also did not do the entire solution like this, just parts where I had to figure out what the words/math was, so this transcribed solution could be wrong and different from the solution in the aforementioned file/pdf.

Anyways:

29. (E) Since is the principal square root of some quantity, . For , the given equation is equivalent to or The left member is a constant, the right member is an increasing function of , and hence the equation has exactly one solution. We write

Since , we may divide by it to obtain so and

Therefore , and the positive root is , the only solution of the original equation. Therefore, this is also the sum of the real solutions.

As above, we derive , and hence . Squaring both sides, we find that

This is a quartic equation in , and therefore not easy to solve; but it is only quadratic in , namely

Solving this by the quadratic formula, we find that [We took the positive square root since ; indeed .]

Now we have a quadratic equation for , namely which we solve as in the previous solution.

*Note*: One might notice that when , the solution of the original equation is . This eliminates all choices except (E).

-- OliverA