The title of this section is maybe a little misleading. The title seems to imply that we’re going to
look at equations that involve any radicals.
However, we are going to restrict ourselves to equations involving
square roots. The techniques we are
going to apply here can be used to solve equations with other radicals, however
the work is usually significantly messier than when dealing with square
roots. Therefore, we will work only with
square roots in this section.
Before proceeding it should be mentioned as well that in
some Algebra textbooks you will find this section in with the equations
reducible to quadratic form material. The reason is that we will in fact end up
solving a quadratic equation in most cases.
However, the approach is significantly different and so we’re going to
separate the two topics into different sections in this course.
It is usually best to see how these work with an example.
Example 1 Solve
.
Solution
In this equation the basic problem is the square
root. If that weren’t there we could
do the problem. The whole process that
we’re going to go through here is set up to eliminate the square root. However, as we will see, the steps that
we’re going to take can actually cause problems for us. So, let’s see how this all works.
Let’s notice that if we just square both sides we can make
the square root go away. Let’s do that
and see what happens.
Upon squaring both sides we see that we get a factorable
quadratic equation that gives us two solutions and .
Now, for no apparent reason, let’s do something that we
haven’t actually done since the section on solving linear equations. Let’s check our answers. Remember as well that we need to check the
answers in the original equation! That
is very important.
Let’s first check
So is a solution. Now let’s check .
We have a problem.
Recall that square roots are ALWAYS positive and so does not work in the original equation. One possibility here is that we made a
mistake somewhere. We can go back and
look however and we’ll quickly see that we haven’t made a mistake.
So, what is the deal?
Remember that our first step in the solution process was to square
both sides. Notice that if we plug into the quadratic we solved it would in
fact be a solution to that. When we
squared both sides of the equation we actually changed the equation and in
the process introduced a solution that is not a solution to the original
equation.
With these problems it is vitally important that you check
your solutions as this will often happen.
When this does we only take the values that are actual solutions to
the original equation.
So, the original equation had a single solution .

Now, as this example has shown us, we have to be very
careful in solving these equations. When
we solve the quadratic we will get two solutions and it is possible both of
these, one of these, or none of these values to be solutions to the original
equation. The only way to know is to
check your solutions!
Let’s work a couple more examples that are a little more
difficult.
Example 2 Solve
each of the following equations.
(a) [Solution]
(b) [Solution]
(c) [Solution]
Solution
(a)
In this case let’s notice that if we just square both
sides we’re going to have problems.
Before discussing the problem we’ve got here let’s make
sure you can do the squaring that we did above since it will show up on
occasion. All that we did here was use
the formula
with and . You will need to be able to do these
because while this may not have worked here we will need to this kind of work
in the next set of problems.
Now, just what is the problem with this? Well recall that the point behind squaring
both sides in the first problem was to eliminate the square root. We haven’t done that. There is still a square root in the problem
and we’ve made the remainder of the problem messier as well.
So, what we’re going to need to do here is make sure that
we’ve got a square root all by itself on one side of the equation before
squaring. Once that is done we can
square both sides and the square root really will disappear.
Here is the correct way to do this problem.
As with the first example we will need to make sure and
check both of these solutions. Again,
make sure that you check in the original equation. Once we’ve square both sides we’ve changed
the problem and so checking there won’t do us any good. In fact checking there could well lead us
into trouble.
First .
So, that is a solution.
Now .
So, as with the first example we worked there is in fact a
single solution to the original equation, .
[Return to Problems]
(b)
Okay, so we will again need to get the square root on one
side by itself before squaring both sides.
So, we have a double root this time. Let’s check it to see if it really is a
solution to the original equation.
So, isn’t a solution to the original
equation. Since this was the only
possible solution, this means that there are no solutions to the original equation. This doesn’t happen too often, but it does
happen so don’t be surprised by it when it does.
[Return to Problems]
(c)
This one will work the same as the previous two.
Let’s check these possible solutions start with .
So, that’s was a solution.
Now let’s check .
This was also a solution.
So, in this case we’ve now seen an example where both
possible solutions are in fact solutions to the original equation as well.
[Return to Problems]

So, as we’ve seen in the previous set of examples once we
get our list of possible solutions anywhere from none to all of them can be
solutions to the original equation.
Always remember to check your answers!
Okay, let’s work one more set of examples that have an added
complexity to them. To this point all
the equations that we’ve looked at have had a single square root in them. However, there can be more than one square
root in these equations. The next set of
examples is designed to show us how to deal with these kinds of problems.
Example 3 Solve
each of the following equations.
(a) [Solution]
(b) [Solution]
Solution
In both of these there are two square roots in the
problem. We will work these in
basically the same manner however. The
first step is to get one of the square roots by itself on one side of the
equation then square both sides. At
this point the process is different so we’ll see how to proceed from this
point once we reach it in the first example.
(a)
So, the first thing to do is get one of the square roots
by itself. It doesn’t matter which one
we get by itself. We’ll end up the
same solution(s) in the end.
Now, we still have a square root in the problem, but we
have managed to eliminate one of them.
Not only that, but what we’ve got left here is identical to the
examples we worked in the first part of this section. Therefore, we will continue now work this
problem as we did in the previous sets of examples.
Now, let’s check both possible solutions in the original
equation. We’ll start with
So, the one is a solution.
Now let’s check .
So, they are both solutions to the original equation.
[Return to Problems]
(b)
In this case we’ve already got a square root on one side
by itself so we can go straight to squaring both sides.
Next, get the remaining square root back on one side by
itself and square both sides again.
Now check both possible solutions starting with .
So, that wasn’t a solution. Now let’s check .
It looks like in this case we’ve got a single solution, .
[Return to Problems]

So, when there is more than one square root in the problem
we are again faced with the task of checking our possible solutions. It is possible that anywhere from none to all
of the possible solutions will in fact be solutions and the only way to know
for sure is to check them in the original equation.