Solving Harder Absolute-Value Equations

The absolute value is isolated on the left-hand side, so I can drop the bars and split the equation into its two cases:

So if the quadratic is positive, there are two solution values to the equation. What if the quadratic is negative?

There are no factors of +2 that sum to −4, so this quadratic does not factor. To solve, I'll have to apply the Quadratic Formula:

So the "minus" case has two solutions, too. Then my complete answer is:

This equation has one absolute-value expression nested inside another one. I'll work from the outside in, splitting things into their cases. First, suppose the argument of the outer bars is negative (that is, do the "minus" case):

I have to stop at this point, because that last line above says that an absolute value is equal to a negative number. But absolute values are never negative! So this case doesn't work; it will provide no valid solutions to the original equation.

Now I'll consider the case where the argument of the outer bars is positive (that is, I'll do the "plus" case):

This equation is valid, and I've got the remaining absolute value isolated, so now I'll consider the two cases again. First, I'll look at the "minus" case:

(It's okay to get a "minus" value for the variable; the only restriction is that I can't get a "minus" value for an absolute-value expression.) Now for the "plus" case:

Then my complete answer is:

This looks awful, but nested absolute values aren't actually that bad. I just have to work in steps. I'll start by taking the outermost absolute-value bars off. Since the original equation is "equals zero", I don't even have to bother with considering the "plus" and "minus" cases:

Now I need to start considering cases. I'll start with the "minus" for the equation above.

This is another "equals zero", so I can go straight to the next set of bars:

Now I'll split into cases again; I'll start with the "minus" case:

Okay; that's one solution. Now I'll consider the "plus" case:

That's another solution.

I'd found these two solutions by taking the first "minus" case, and tracing it down to its end. That consideration started back at the purple "Now", above. So I need to start there, and consider the "plus" case.

I'll split this into two cases; I'll consider the "minus" case first:

This case doesn't work, because it's telling me that an absolute value is equal to a negative number. That can't be true. So I discard this branch and try the "plus" case:

This works, so I can proceed to the two branches for this equation. I'll do the "minus" case first:

Now I'll do the "plus" case:

I've considered every case, and have arrived at four solutions. My complete answer is: