Question

How do you solve `-sqrt(8x+4/3)=sqrt(2x+1/3)`?

Answer 1

`x=-1/6`

Explanation:

At first glance the problem seems to have no solution, since the left hand side is negative, while the right hand is positive. Thankfully, there is one number that fits this seemingly contradictory requirement - and that is zero.

Both sides vanish for `x=-1/6` which is the root.

Note that you could try finding the solution by the standard method of squaring both sides. In this case, squaring gives `8x+4/3=2x+1/3` which leads to `x=-1/6`. However, remember that squaring can lead to extraneous roots so that you must always check whether the solution you find this way actually satisfies the original equation.

To give an example of the kind of trouble squaring can get you into, consider the similar equation `- sqrt{x+1} = sqrt{2x-1}`. Squaring both sides will give you `x+1=2x-1` or `x=2`. You can check that this ~does not~ satisfy the original equation.