>> This is Part 1 of the square root property of equations.
This is the property we're using.
And we do the following six problems on this video.
In this video, we're going to go
over what's called the square root property of equations.
And we're going to be using it to solve quadratic equations.
But, first, let's remember how
to solve a quadratic equation by factoring.
For instance, X squared equals 25.
Remember, we set the equation equal to 0 and then we factor.
X squared minus 25 is the difference of two squares,
so it's X plus 5 times X minus 5 equals 0.
And then we set each factor equal to 0.
So either X plus 5 equals 0 or X minus 5 can be equal to 0.
So we've got X equals negative 5 or X equals 5.
So our solutions are negative 5 or 5.
Now, you could check that in the original equation right here.
If you plug in negative 5 for X,
negative 5 times negative 5 is 25.
And if you put in 5 for X, 5 squared is also 25.
So notice for X squared equals 25, we get two solutions.
And usually when we're solving quadratic equations,
we get two solutions.
Now, we're going to try solving this a different way using
what's called the square root property of equations.
So I'm going to start with the same equation,
X squared equals 25.
Now, keep in mind, we know the answers are going
to be negative 5 and 5.
There isn't one solution.
A lot of people think, well, I'll just take the square root
of both sides and I'll get X equals 5.
So, in other words, they'll say, Well,
I'll just take the square root of both sides.
And then I'll write square root of X squared is X
and the square root of 25 is 5.
But you'll notice that only gives you one solution.
The trick is, when you take the square root of both sides
like this, on your next step when you're solving for X,
you have to put a plus or minus in front
of whatever the square root of 25 is.
Now, in this case, you could simplify it to 5.
But if you left it as the square root of 25, you would have plus
or minus square root of 25.
This means there's actually two solutions: X equals 5
or X equals negative 5.
And you get right back to your same solution
of negative 5 or 5.
Let's try another quick problem,
and then we'll state the property.
How about if I had X squared equals 7.
So, hmm. Well, how about if I just say, well, I'm just going
to take the square root of both sides.
And then my next step will be X equals --
remember, you had to put a plus or minus in front
of that number, plus or minus the square root of 7.
So there are two solutions here.
X can be the square root of 7 or negative the square root of 7.
In this particular problem,
the original X squared equals 7 could not have been solved
by factoring.
All right.
So let's state the property now.
Here is the square root property of equations.
If A squared equals B, then A is plus or minus the square root
of B. So that's what we just did.
Let's do a problem again.
So if I have something squared equals some number, like 9,
right, we've got the X squared, I could write
that that's the same as X equals plus
or minus the square root of 9.
And, then, square root of 9 is 3.
So I could simplify that a little further and put plus
or minus 3, which gives two solutions: 3 or negative 3.
Now, if it's easier on your original problem here,
what you go do when you see X squared equals 9,
you could just put a square root over each number.
That's okay.
But remember, at the next step, you're going to have
to put a plus or minus in front of that square root of 9.
And, in fact, you could skip down to this third step,
X equals plus or minus 3.
Be careful when you use this property.
You have to begin with just something squared equals
a number.
It can't have 5X squared.
It can't have X squared plus 9.
It just has to be something squared.
So now let's work a few more problems.
Right. Put the video on pause
and try this one, X squared equals 13.
All right.
If you'd like, you can put the square root to begin with.
That's up to you.
But to use the square root property, that means X equals --
and you must remember to put plus
or minus the square root of 13.
Now, you can't simplify the square root of 13,
so this gives you two solutions.
X can either be the square root of 13
or negative square root of 13.
Now, I haven't been checking my answers so far
because they're fairly obvious.
But let's go ahead and check that negative square root
of 13 is a solution in the original, X squared equals 13.
So if you're going to plug in negative square root of 13
for X, remember you have to put that in parentheses
to show you're squaring it.
And, then, if you're squaring a negative,
it's going to be positive, and square root
of 13 times square root of 13 is just 13.
So you can always check your answers.
And you can do the same thing to check that square root
of 13 is also a correct solution.
All right.
Here's another example.
X squared minus 6 equals 10.
Now, notice you don't have just something squared
on one side of the equation.
You have X squared minus 6.
You must isolate that thing that's squared first all
by itself.
So we first want to add 6 to both sides
to get X squared equals 16.
Now we've got it in a form
where we can use the square root property.
So, at this point, if you want,
you can put the square root over each.
That's up to you.
But the next step, remember, once you change it
from X squared to just an X, right,
when you take the square root, you have to put plus or minus
in front of that square root.
So plus or minus 4.
So this gives us two solutions, right?
X could be 4 or it could be negative 4.
So let's go ahead and just check -- let's check 4.
Check X equals 4 into the original,
which is X squared minus 6 equals 10.
So you're going to plug in 4 for X. All right.
So on the left side I've got 4 squared minus 6.
Remember you do exponents first.
16 minus 6 is 10.
And 4 checked.
Now, remember, when you check negative 4, you're going to have
to put that negative 4 in parentheses
because negative 4 squared is negative 4 times negative 4.
You'll get exactly the same check
after the first step, 16 minus 16 is 10.
So that means negative 4
and positive 4 are the solutions to this one.
All right.
Let's try this one.
3X squared minus 1 equals 8.
Well, the X squared isn't isolated.
And I have a 3X squared.
Plus I also have got this minus 1.
So first I need to add 1 to both sides
to get the 3X squared by itself.
And notice the X squared and not isolated.
I have 3X squared.
So I also have to divide by 3.
So I have X squared equals 3.
And now, finally, I can use the square root property
at this point.
I can write the square root of both sides if that helps you.
So I have X equals.
Remember, always remember to put plus or minus.
So X is plus or minus the square root of 3,
so I have the square root of 3 and negative square roots of 3.
And let's check this problem.
Now, to check, you would really have to do the check twice.
You could check negative square roots of 3
and also positive square roots of 3.
I'm going to do one of them.
So I'm going to plug in negative square roots of 3 for X.
So I have 3 and then, for X,
I've got negative square roots of 3 squared minus 1.
Now, the order of operation,
remember you do the squaring first.
So I'm going to do negative square roots of 3 squared,
and that is going to just be positive 3.
Because the negative times a negative will be positive,
and square roots of 3 squared is 3.
And so that gives me, order of operations, again,
do multiplication before subtraction,
9 minus one is 8, 8 on both sides.
So negative square roots of 3 is the solution.
When you plug in square roots of 3,
you'll also get that that checks.
All right.
There are -- we're going to do some different kinds
of problems on the next video.
A little preview is, what if you had something like this,
X minus 5 squared equals 9.
We'll do this on Part 2.
How are we going to deal with those?
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