 Hi guys! I'm Nancy.
I'm going to show you how to solve by completing the square.
What is completing the square?
It means getting your equation into a neat
perfect square form like this.
Where you have x plus or minus a number squared equals another number.
Why would you want to solve this way?
Good question. It's not the best way usually.
It's usually easier or faster to solve by factoring
if possible.
Or quadratic formula. Which you can always use.
But probably you've been told
that you have to solve by completing the square.
So you need to know how to do it.
So I will show you the easiest way to solve by completing the square.
Unfortunately, you're not given this form straight away
you have to get it.
Usually you're given one of these forms.
OK, these are the cases that I'm going to show you.
One case is if you have just x^2.
Positive x^2 as your highest term.
Another case is if you have a coefficient greater than 1 in front of your x^2.
Like 2x^2, 3x^2, etc.
Third case is if you have a negative coefficient in front of x^2.
-x^2, etc.
And this case I put here to show you that
if you get something like this that doesn't have
an x term, a middle term,
you can't complete the square.
Sometimes people get confused about that.
So really I'm going to show you these 3 cases.
OK. Say you have an equation like this
that starts with x^2. Positive x^2.
The first step, actually for all of these completing the square problems.
The first step is to move this constant to the right.
So in this equation the way you would do that
is to add 7 to both sides.
So always just move that constant over to the right.
So what you get is: x^2 + 6x = (positive) 7.
Notice that the sign flipped. That will always happen.
If it was negative here, it will be positive there.
If it was positive here, it would be negative on the right side.
The next step
is to add a number to both sides.
So you're going to have plus something and plus something.
What number do you add?
You always get the number by taking
the coefficient of x, the coefficient of the middle term,
in this case 6.
Dividing it by 2 and squaring it.
And that will be what you'll add to the left and right side.
So in this case it's: (6/2)^2
Which simplifies to 3^2.
Which equals 9.
So we'll be adding 9 to the left side and right side.
So here we fill in + 9.
We've added the same thing to both sides.
What you do to one side you must do to the other.
So that we don't change the overall value of the equation.
So that it's balanced.
OK, so in the next line that simplifies to
x^2 + 6x + 9
And on the right side we can combine these numbers into 16.
7 + 9 = 16
Why did we do that? Why add a number to both sides?
The only reason is so that immediately after
we can write the left side as a perfect square.
x plus or minus some number squared = 16.
The number you write here is always
half of what the middle coefficient was.
So in this problem is was 6/2.
Positive 3.
If this were a negative 6, you would put negative 3.
So always in the perfect square side
use half of what the middle coefficient was.
So now we have: (x + 3)^2 = 16
That's great you've completed the square.
You have something that will be very easy to solve now.
And the whole point of doing that was so that you can
square root both sides in order to solve.
We can literally draw a big square root on the left side and a square root on the right side.
Be careful! Because when you square root
a constant on one side, you actually get two solutions.
Positive and negative.
Very important. And many people forget this.
So try to remember when you square root the side that has the constant
you immediately have two solutions possible.
The positive and the negative version.
OK, on the left side meanwhile
what that does is get rid of the square.
So you just have: x + 3 (what was inside the parenthesis)
equals positive and negative square root of 16.
You probably know, square root of 16 is 4.
Since 16 is a perfect square.
So then in the next line you can write
x + 3 = (plus and minus) 4
The easiest thing now is to separate this into two solutions.
You're going to solve for two different answers.
One equation you can write
using the positive version.
So: x + 3 = 4
And the other one, you'll use the -4 version.
x + 3 = -4
It's very important to separate it out, because you'll end up with two solutions.
So what this gives you using basic algebra
subtracting constants is: x = 4 - 3
Or positive 1.
And on this side: x = -4 - 3
Which is negative 7.
So your two solutions
using completing the square are:
x = 1 and x = -7
OK, say you have an equation that looks like this
with 2x^2 or 3x^2, etc.
where the coefficient of x^2 is something bigger than 1.
The first step will be the same.
You're going to move this constant to the right side.
So since it's -3 right now, you can add 3 to both sides
in order to move it to the right.
And you get: 2x^2 - 10x = (positive) 3
OK. Now what you have to do
because there is a 2 attached to the x^2
(or 3, or 4, etc.)
You have to divide it out from every term in the equation.
That will make it easier for you to complete the square.
It will make it possible for you to complete the square.
So you're dividing every term by 2.
That includes the right hand side.
So what you get is
just x^2 now (which is good)
minus 5x (since it was -10x/2)
and on the right side
You have 3/2, which is already in simplest form.
It's just 3/2.
On the right side.
The next step is to add a number to both sides.
So you're going to add something here
and the same something here.
What number do you add?
You add...
this coefficient of x
but divided by 2 and squared.
So it's always
the coefficient of x over 2 then squared.
And that is what you add to the left side and the right side.
So in this problem that's:
(-5/2)^2
which simplifies to
positive 25
(because -5 * -5 is +25)
over 2^2 on the bottom. Which is 4.
So you're adding the fraction 25/4 to both sides.
I gave you one that has a fraction
because a lot of time completing the square problems
end up a little ugly and have fractions involved.
So you'll probably have one like that.
Now you added the same number to both sides
because you don't want to change the overall value of the equation.
You want to keep it balanced.
And what this lets you do immediately after
is write the left hand side as a perfect square.
Meaning x plus or minus a number squared.
That's the whole point of adding that number.
Is so that on the left side you can immediately write:
x and (some number here) squared.
What number do you write?
It's this coefficient over 2.
So it's -5/2.
And since it's negative, instead of addition, it's subtraction.
-5/2
squared equals the right hand constant.
Since this is an uglier example it has some fractions involved.
You'll have to get a common denominator in order to combine them.
Or plug them into your calculator, if you're allowed to use a calculator.
A lot of calculators are very good at adding and simplifying fractions.
What you have to do is get a common denominator
which would be 4 in this case.
So...
You have 25/4 and you have 3/2
which can also be written as 6/4.
If you multiply the bottom by 2 to get 4.
You also have to multiply the top by 2 to get 6.
So in the next line
you can rewrite this as (x - 5/2)^2 =
and combine the fractions.
Since the denominator is the same. You keep it as 4.
And the numerators you add. So 6 + 25 gives you 31.
Great! So you've completed the square
and this is in a form that would be very easy for you to solve.
What you want to do is square root both sides.
So on the left side it's a big square root.
On the right side you do a square root of the constant,
but you immediately get positive and negative versions.
So remember when you square root the side that is just a number
you have to put plus and minus.
If you don't you'll only have half the solutions.
You'll only have 1 instead of 2 solutions in the end.
So it's very important. People forget it all the time.
But, you want to square root both sides
what that leaves you on the left
is just x - 5/2.
Because your square root undid the square
and you just have what's left inside the parentheses.
So: x - 5/2 =
plus and minus square root of 31
over square root of 4. Which you can just write as 2.
Then you'll want to use some simple algebra
to move this constant over, so you have x alone.
And your 2 solutions will be:
x = 5/2
plus or minus root 31 over 2.
You can actually combine this into one fraction
which most people would consider simpler.
So you may want to do that.
And your final answer would look like this
x = 5 plus or minus square root 31
as your numerator. All over 2 as your denominator.
So these are your 2 solutions
you got from completing the square.
It may not look like it, but this is 2 solutions
because you have 2 versions.
You have (5 + square root of 31) / 2.
And you have (5 - square root of 31) / 2.
So those are your answers.
OK, say you have an equation that look like this
that starts with a negative x^2.
Or -2x^2.
Where the coefficient of x^2 is a negative number.
Your first step will still always be
to move the constant over to the right-hand side.
So the next line you would have:
-x^2 - 6x = -7
Notice the sign changed.
And then...
you don't want the negative in front of the x^2
before you complete the square.
So you actually will divide every term by a -1
to get rid of it.
Since there actually is a -1 here. You don't see the 1.
but you can divide every term by -1.
Even the right hand side.
What that does is it flips the sign of every term.
So instead of -x^2, you'll have positive x^2.
Which you want.
(positive) 6x
and on the right side, positive 7.
And then your steps from there
will be the same as in problem 1 and 2.
Adding a number to both sides that you get from
dividing this number by 2 and squaring it.
So you would add 9 to both sides.
(6/2)^2. 9 to both sides.
Writing it as a perfect square.
Solving by square rooting both sides.
From then on it would be the same.
So notice this is a lot like
case number 2, example number 2,
where you divide out anything that was in front of x^2.
OK, things that tend to trip people up
about completing the square.
The biggest thing is...
forgetting to put plus and minus in the solution.
So you if you get to the point
 where you have an equation that looks like...
this.
When you square root both sides to solve
a lot of people forget to put plus and minus here
on the constant side
and just write the root.
Lots of people forget that.
You don't want to miss the other solution.
There are 2 solutions
and if you don't put the plus and minus
you're only getting the positive solution.
So remember the plus and minus.
A lot of people forget that.
So you just have x - 4 = plus and minus 5. In this case.
Then you would keep solving from there.
So that's the biggest thing people forget in completing the square.
Another thing that trips people up
is sometimes they think
you can complete the square
if there's no x term.
For instance if you had
3x^2 - 121 = 0
if that was your equation
you can't complete the square because you have no x term.
Such as 3x^2 + 10x - 121
If there's no x term there
you have no coefficient to use when adding a number to both sides.
So you can't complete the square if it looks like that.
You'll have to use another method.
So... factoring. Quadratic formula.
Something like that.
So I hope that helped you understand
how to solve by completing the square.
Completing the square is fascinating...
No.
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