Let's take a look at solving quadratic equations.
Quadratic equations are degree-two equations.
You'll notice the highest exponent on the
variables is two, which means it's quadratic.
You will need to be able to solve these all throughout the semester.
Let's look at this one: 4 x squared plus 7 x equals 2.
This one is not in standard form, which means
standard form is when you have descending powers
and zero on the right hand side.
So, to get this in standard form we're going to
first subtract 2 from both sides of the equation.
Now it's in standard form.
We want to see if this equation will solve by
factoring, which is one of the techniques that
we'll use a lot to solve quadratic equations.
Not all quadratics factor, but this one does.
Let's look at it.
To solve, to factor,
you're going to have two expressions.
Now, a couple of good notes: I see 4 x squared
here, and I have choices.
I can put 2 x and 2 x, but if I do that, the only
factors of 2 are 2 and 1.
And so then, that means I'd have a 2 x and a 2 in
one of these factors, which means there'd be a
common factor of 2.
Notice there's not a common factor 2 to begin
with, so I know that, by trial and error, it's
not going to be 2 x and 2 x.
I'm going to try 4 x and x, and then
I need 2 and 1.
Well, I want to get a 7 x in the middle, so I want
to get 8 minus 1 to give me the 7 x in the middle
with a plus 8 x and a minus x, so it does factor.
Once I get it factored, I simply set
each factor equal to zero.
So, I'm going to have 4 x minus 1 equals 0 or x
plus 2 equals 0, and I'll solve each of these equations.
Four x equals 1, so one of my solutions is x
equals 1/4, and the other solution is x equals
negative 2, so I end up with two solutions here
to this quadratic equation.
So this is an example of solving by factoring.
Factoring is a very important skill that you'll
need to be able to do throughout the semester, so
you need to practice that and make sure that you can solve by factoring.
Alright, next we're going to look at solving quadratic equations by other techniques.
Let's look at another a quadratic equation.
This one is quantity 3 x minus 2 squared equals 16.
If we expand this out, the first term would be 9 x
squared, so we notice it's quadratic.
But importantly, whenever you see a quadratic in
this form of something squared equals a number,
there is a technique or strategy called the
square root method that works really nice.
If you think about this, you're looking for some
number that squared gives you 16.
Well of course, there's two numbers that you could square and get 16.
There's positive 4 and negative 4.
So the way we write this using the square root
method is 3 x minus 2 equals plus or minus 4.
Again, positive 4 squared gives me 16 and
negative 4 squared gives me 16, so there's always
a plus or minus when using the square root method.
Now, this actually means two separate equations:
3 x minus 2 equals 4 or 3 x minus 2 equals a
negative 4, and then we'll solve each of those equations.
We're going to get on the left 3x equals 6, so x
equals 2. From the right hand side, you're
going to have 3 x equals negative 2, so x equals negative 2/3.
So we end up with these two solutions, and what
you want to remember is whenever your quadratic
is in this form or can easily be put in this form,
something squared equals a number, you want
to think about using the square root method, and
you want to remember not to forget the plus or
minus because you can either square a positive
number and get 16 or you could square a
negative number and get 16.
So there's always a plus or minus that you need to remember.
That's a common thing students forget to put.
Alright, let's look at another quadratic equation.
Let's look at another quadratic equation, this
one: 2 x, parentheses x minus 1, equals 5.
Notice that if we expand it out, we have 2 x
squared, so it's a quadratic equation.
So first, I'm going to multiply on the left hand
side and get 2 x squared minus 2 x, going to go and subtract the 5,
get this quadratic in standard form.
Now it's in standard form, but it doesn't factor
easily, so we have another technique to solve
this quadratic called the quadratic formula.
First, it has to be in standard form, which this one is.
Standard form is when it looks like a x squared
plus b x plus c equals 0. The quadratic
formula is x equals negative b plus or minus the
square root of b squared minus 4 a c,
all of this is over 2 a.
This is a very important formula that you need to remember.
A lot of students forget that the b is over 2a,
the whole numerator is over 2a, so you need to remember this formula.
Notice that we have our quadratic in standard
form: our a is 2, our b is negative 2, and our c is negative 5.
So I'll use this formula, plugging in my numbers.
I'm going to get negative b, which will be a
positive 2, plus or minus the square root. When
you square b, you have negative 2 squared minus
4 a c, and all of this is over 2 a.
Okay, so now we're going to simplify a little bit more.
Underneath the square root, what are we going to have?
Let's see, we're going to have negative 2
squared, which is of course 4, and then notice
that we have two negative signs so that's going
to change to plus, so we'll have 4 plus 40, which
is 44, under the square root. Then in the denominator, we have 4.
Okay, so we've gotten here to 2 plus or minus the
square root of 44 over 4. Now comes the
tricky part for students, simplifying this.
First of all, can we simplify the square root of 44?
Are there any perfect squares in 44?
We need to look for the largest one, which in
this case is 4, so I can first reduce the square root of 44.
It'll be 4 times 11; 4 is the biggest one. So I
write this as 2 square root of 11 over 4, and now
we need to reduce this further.
There's lots of places that students make mistakes.
One of the mistakes they make over here is
they're so tempted to just put 1/2 and forget
about the 4 that goes underneath this term in the
numerator as well, so don't do that. Don't write
1/2 plus the square root of 44.
Now when we get to here, we look at these three
terms, and we look for the greatest common factor
in those three terms, which of course is a 2.
So we're going to reduce each term by a factor of
2. So 2 divided by 2 is 1, and when we do 2 into
this we have the square root of 11,
and 2 into 4 goes 2.
So, our final answer is 1 plus or minus the
square root of 11 over 2. You need to be very
careful on problems like this where you reduce.
A lot of students will just forget about this 1.
They'll just write plus or minus the square root
of 11 over 2 and think that 2 goes into 2 maybe 0
times. But remember there should be two terms in the answer
if there was two terms up here, then there must be two here.
So this, of course, can also be written 1 plus
the square root of 11 over 2 and then 1 minus the square root of 11 over 2.
Be very careful when reducing on these types
of quadratics. You want to practice these a lot so that you get this.
This is one of the problems that students missed a lot on one of the tests.
Alright, let's look at another quadratic equation.
Let's look at this example here: 4 plus 1 over x minus 1 over x squared.
Notice it's another rational equation, remember I
said rational equations you have fractions.
So we've got variables in the denominator.
First thing we want to do is to clear fractions.
We want to multiply by the least common
denominator, which will be x squared.
So if we do that, we'll get 4 x squared plus x minus 1 equals 0.
So notice that it's now a quadratic equation, and
we want to solve it, and again it doesn't factor
easily, so we'll again use the quadratic formula.
Do you remember the quadratic formula?
Think about it for a minute and write it down.
It's in standard form, so we're going to use the quadratic formula.
And we're going to get negative b, b is 1, so we
get negative 1 plus or minus the square root of b
squared is 1 minus 4 times a, which is 4, times c which is a negative 1,
and all of this is over 2 a.
So it's very important that you remember the quadratic formula.
Let's see what we get under the radical.
We have one and again there's two negative signs
so it's plus 16, 1 plus 16 is 17, and in the denominator we get 8.
Now this particular answer doesn't simplify, the
square root of 17 can't be simplified,
so we are finished now.
Our solutions are negative 1 plus the square root
of 17 over 8, and the other solution is negative
1 minus the square root of 17 over 8.
So again, think about the different quadratics,
think about which method works best.
Sometimes factoring works great, sometimes you
want to use the square root method, and sometimes
you want to use the quadratic formula.
All of those techniques are very important.
First thing, remember you need to identify and recognize equations.
This is quadratic because it's degree-two, and
think there's three different methods we talked
about: factoring, square root method, and the
quadratic formula. You need to practice on
all of these until you are successful with each technique.