Now we're going to solve quadratic equations using the quadratic formula. This is the third way that I've got
algebraically to solve quadratic equations.
So first thing we have to do make sure your quadratic equation is in standard form in
descending order.  I usually have my students indicate what the value of "a", the value of "b". and the value of "c".
And then here's your formula. You should know this.
I always get my students to go find the discriminant, because the discriminant gives you a lot of
information.  The discriminant is just the number
under the radical, does not include the radical, just the b squared minus 4ac.
Now if you find your discriminant if it's positive,
then I'm going to have two solutions.
But now depending on what number:  I have if I have a perfect square
I have two rational and all rational means base word here is ratio.
I can write the answer as a fraction and I get an exact value for it.  Now if it's not a perfect square,
then I've got two irrational which means not rational solutions. There's no way I can write a fraction or
write the value into a fraction that will give me that exact answer back once I apply the
radical.  Okay.  Now if my discriminant is zero I have one rational
solution, because 0 is a perfect square.
If my discriminant is negative. I have no real solutions.
It is an imaginary number, but we're just going to say no real solutions in this class.
All right, let me keep on going I've got some examples.
You should have seen the quadratic formula, and you should know the quadratic formula.
Now this - I don't know what you did in the past with respect to the discriminant,
but you need to know what happens when you have a positive, a zero, or a negative.
So I've got some examples. We're going to look at - so first thing
I want you to find on these is the value of the discriminant.  So all of these are already in standard
form so you've already got that done.
So I need you to go find the discriminate and see if you come up with what I did.
Pause me and go find it. Then come back. You should have enough room. You can find it.
So on this one if you find your discriminate, you should have ended up with one.
In this one our discriminate should be zero.
Discriminant here hopefully you came up with negative 4 and my discriminant down here I came up with 17.
So I have an example of every one of the situations
indicated earlier.
One is a perfect square.
Move this up just a little bit.  One is a perfect square, so I'm going to have two rational solutions.
0, here  is 0 right here in the middle.  One rational solution.
Negative 4, right here, it is an imaginary number
or some imaginary solution and
17
positive, but it's not a perfect square, so I've got two irrational solutions.
I thought I would highlight that.  Now my solutions - here comes the quadratic formula.
So what I need you to do is go find your solutions.
Indicate, you know, using your quadratic formula;  what they are and then I want you to see if you come up with what I do.
In this one I came up with x = -2 or x = -3.
Zero - you should have ended up with x =7.
This is when I have a double root.
So these two, actually one and two, if you come up with a perfect
square; the thing could have been factored to start with.
In this one
no real solution. I really like this one. It's very fast, but you won't get any,
usually.
Now in this one.
Let's see if you first ended up with - you should have ended up with 11 plus or minus square root of 17
over 4.  This is an exact answer.
So let's see if I can go and get my approximations.
So because this is a binomial I need parentheses, so I've got 11.
I do the plus route - the upper route.
Square root 17,
now I had to end the
parentheses here with the 17 to tell the calculator that this is under the radical,
but I need another parentheses to tell the calculator
this is my numerator, so however many left-handed parentheses you have that's how many right-handed parentheses
you need.  So let me divide by 4.
So there's my
3.781, if I round to the third decimal place.
The only difference between these two problems is that minus - so if you'll do your repeat,
cursor back in,
change that to a negative, hit enter, you do not have to cursor out.  Okay?
So if I go on down to solve a quadratic equation for exact -
exact solutions that means algebraically. If it's in the form of ax squared = p or
you have a quantity squared equals p.
Solve for the square and remember to take square roots.
When it's not in the form of one; write it in standard form.  So if it's not in this form rewrite it in Standard form.
Okay?
Try to factor - usually factoring is faster. If you can notice it and use the principle of zero products.
If it's not possible to factor or factoring seems really difficult
use the quadratic formula.
The solutions of a quadratic equation can always be found using the quadratic formula.
But it has to be in this form first, so you've got some problems to do.  So get back in touch with the quadratic formula.
Talk to you later.  Bye-bye.
