Hello class welcome to unit 4 video 7 the quadratic formula
So today you're going to learn how to solve quadratic equations by using the quadratic formula
So let's just look at an example
competitors in the 10-meter Platform diving
competition jump upward and Outward before diving into the pool
The height of a diver in meters above the pool after two seconds can be approximated by the following equation
H is equal to negative 4. 9 t squared plus 3t plus 10, okay?
It says how many seconds it take for the diver to hit the pool the first thing
I would notice is it's a quadratic because my leading term is
A t squared the next thing I would look at is I see that there's decimals, okay?
So what I would do is I would say could I factor this
so if we're factoring
Can we solve this one by factoring?
Maybe but with that 4.9
It just makes things really nasty and I wouldn't even want to try so factoring is out. That's a no okay
so
No
Could we solve by graphing?
Yes, we could okay if we graph the function. We could find it. So yes, we could solve by graphing
Okay, so we could solve by graphing on this one, and I will show you how to do that in a minute our
Last resort, then is if you maybe don't have a graphing calculator, and you can't factor it is to use the quadratic formula
so just to remember what the quadratic formula remember you have the opposite of B plus or minus the
square root of
B squared minus 4ac
all over
2a so in this case my a is equal to negative
4.9 my b is equal to 3 so what's in front of the t and my C is equal to 10?
So if I plug this one in I would have negative b so negative 3
Plus or minus the square root of my b. Squared so 3 squared
Minus 4 times my a negative 4.9 times my C. Which is 10
Square root of all that all over 2 times a which is negative
4.9 I
Would recommend simplifying what is underneath the radical first?
Okay, so I would take out your calculator and do that
So sorry mine is just loading here
So if you don't have a calculator, I would recommend grabbing it well mines loading up
And if you do have it start typing in what's underneath the radical sign?
So that would be this 3 Squared minus 4 times negative 4 times 10
all right, so
Looking at my calculator. I'm going to type in what is underneath
my radical
right here, so
parentheses 3 and parentheses squared
minus
4 times
Negative 4.9
Times 10 and
Hit enter okay. So what's underneath there is a two thou.. 205
so this simplifies
So I would have x is equal to negative 3 plus or minus the square root of 2 05
2 times negative 4.9. You can type it in or its negative 9.8
Okay, then define my answer is notice. How is a plus or minus?
I have 2 it's going to be x is equal to negative 3 plus the square root of 205
negative 9.8, and
X is Equal to Negative 3 minus square root of 205
Negative 9.8. Okay, so let's go ahead and do that
So on our calculator
Make sure you do negative use the negative below the three so negative 3 plus
second square root 205 and I would hit enter and then
Take that answer so divided by negative 9.8
So this gives me that x is equal to negative
1.58
1.55
So my answer for this one is
negative one point
Sorry
155 so what is x really representing if I look up x is well in this case. It's actually t
But it's the same thing as x so t is the time can I have negative 1.155 seconds?
No, so that is not the answer we are looking for in this situation. So I try my next one I
have
Negative 3 minus the square root of 205 I
would hit enter
Divide by negative
9.8
enter and I get 1. 767
So for here x is equal to negative
Or sorry
Not negative 1 point really need the answer 1.767  seconds so with that one work
Yes, okay?
so that's how long it will take for the diver to hit the pool 1.767 seconds and
Just to remind you how to do this one on your graphing calculator
You would go if you wanted to graph it to your y equals
Clear out any equations you have and type it in so you'd have negative 4.9  x squared
Plus 3x
plus
10 I'm going to make sure my window is to standard so Zoom 6
I'm going ahead and hit graph
Okay, so there is my equation. I'm sorry
There's my parabola being wrapped and if I want to find this value to tape it on right?
Oh right, here made a mistake
um
Forgot to put in the plus sign
so second if you ever need to insert anything Ins is insert a
Plus alright, let's look at that again
Now it looks like a parabola
Here's where my negative answer I'm not concerned about a negative time, so I'm not worried about that
I need to find this one so remember to find it you go second calc
Which is above trace that is a zero value which is number two?
To find my zero I have to be to the left of it
So I'm going to just bring my little pointer over here. See how it's flashing to the left at this point. Yes, so enter
Come down enter and enter
So it says 1.767  which is what we got for our answer
So if you want it to graph it, you could as well or you can use the quadratic equation quadratic formula
So moving on quadratic formula
Okay, the solution of a quadratic equation in the form ax squared plus Bx plus C equals 0 it's where a doesn't equal 0 can
be given by the following equation
So this right here is the quadratic formula. Do you need to know it? Yes, okay?
So I won't be giving you that on your test
In the quadratic formula is something we call the discriminant the cosine is called the discriminant
And what the discriminate it does is it tells you how many solutions you have we've already went through how many solutions you can have
But if my discriminant is greater than zero what that means is if it's positive
Okay, I'm going to have two answers the reason is I'm going to have negative B plus or minus some positive number
Okay over two times a so that's going to give me two answers okay, and here. They are here
So this one will be two
real solutions
Now it tells me if my square root is equal to 0 okay well, let's think about it
If I have negative B plus or minus, what's the square root of 0?
Square root of 0 is going to be 0 if I add or subtract the zero
It's going to give me the same answer, so this case only going to give me one solution
one Real Solution
and
The next one if my discriminant is negative
less Than 0
Okay, I will have it will not touch this will be no real
Solutions however
How many complex solutions what I have I would have two complex
So there's no real so graphically it won't ever touch the x-Axis, but there could be two
Complex solutions thinking about that Imaginary I
Ok so that's when you'd probably have you'd include that
You know I is equal to square root of negative [one], so there's no real that means. It doesn't touch the x-Axis
But there would be two complex solutions
Moving on to example one
You need to know the quadratic formula
You can want there's plenty of songs online that will help you most of them. Go to row row row your boat
so
solving using the quadratic formula
Same as with solving any quadratic. You need to make sure you have x
Everything onto one side, so this one will be x squared minus 12x
Subtract 28 is equal to 0
[okay], my a is. What's in front of my x squared which is one my b
Is what's in front of just x so negative 12 and my C is the number by itself So negative 28?
After I have that information, okay, I'm going to go ahead and plug it into the quadratic formula
So x equals negative b. So this would be good positive 12 plus or minus the square root?
negative 12 squared minus 4 times 1 times
negative 28
Sorry that got really crowded all over 2 times my a which is 1
So x is equal to let's get out our calculator and type in what's underneath the radical
Especially when you have a negative, so make sure you use parentheses
So we're going to do parentheses negative 12 and the parenthesis squared
Minus, it's four
Times our a which is one, so I'm not going to write that one times our c. Which is negative 28
Ok this is what belongs underneath my radical it's 256
So x is equal to 12 plus or minus the square root of 256
over 2 okay 256
Hmm sounds to me like it might be a perfect square. Let's try it second square root of
256 it
is so I would have
12 plus or minus 16 2 so let's type that in I
would have 12
plus 16 hit enter divided
2 so that means my first answer is
14 my other one is
12-16
Which we know is negative 4 and then we're going to divide that by 2 which gives me negative 2 so my two
Answers for this one is x is equal to negative 2 and positive 14 I
Had nice anthers like this no decimals
Nothing that so that brings me to believe that maybe I could have just factored this let's just think about it
What multiplies to negative 28 but adds to negative 12?
Well 2 times
14 is 28 so if I have a negative 14 and a positive 2
They would add to negative 12 and if I solve I will get x is equal to 14 and x is equal to negative 2 So that works the directions did say solve using the quadratic formula
So that's what I had to do
But if you are given the option notice how much quicker factoring can be so really having your factoring skills up to par
Will help speed things up for you
All right
Example 2
It still says to solve using quadratic. So we're going to do that. I look at it
Everything is already on to one side
So my a for this one is 2 my b is 4 my C is negative 5 so x is equal to
Opposite of b so negative 4 plus or minus square root B squared so 4 squared
minus 4 times my a times my c
all over 2 times a
So 2 times 2 okay on my calculator. I can type in what is underneath my radical
I also know that 4 squared is 16, so I'm going to do 16 minus 4 times
2 times negative 5 in some of this you do in your heads you don't need to get out your calculator. So whoops
What did I do wrong?
Check it out go to oh I use the wrong negative
4 times 2 I forgot times
Negative 5
There we go
That's 56. I would try square root of 56. It's not a perfect square, but if you didn't know you could try it
So I
Am left with right now
X  is Equal to negative 4 plus or minus the square root of 56 and
All over 2 times 2 which is 4 the next thing. I would do is I need to simplify my radical 56
What's my perfect square in 56?
4 x 14 is 56 so it's really square root of 4 x 14
So I have negative 4 plus or minus the square root
I'm going to actually write it all so this would simplify to this would be 2 square root of 14 all over 40
K and you can leave your answer like this, but the one thing I would notice is are both of these pieces
Divisible by something that this is so these are all divisible by 2 so I'm going to go ahead and do that
So my final answer would be negative 2 plus or minus 2 divided by 2 is one so square root of 14?
All over 2 and that would be my final answer
And that's how I would want it simplified
It was a real life problem. I'd want you to type it in and get me or to get me the two answers
See so using the quadratic formula. I notice everything is on the one on the same side
So I would say a is equal to 3 b is equal to 5 C is equal to 4
plugging it in I have x is equal to the opposite be so negative 5 plus or minus square root of
5 squared minus 4 times my a which is three
Times my C. Which is 4 all over 2 times my a which is 3
I'm going to save some time you can type it in if you want
But underneath my radical if you type that in you get the square root of negative 23 all over 6
Simplify my radical
Square root of negative 23 I know it's negative so I'm going to write it as negative 1 times 23 23 is a prime
So I can't break it down any further. So my final answer will be x is equal to negative 5 plus or minus I
Square root of 23
6 and there's no more simplifying that can be done for that one okay, so that is
Examples 1 2 & 3 using the quadratic formula if you have any questions on that
Please let me know if you'd like a break before we get to the word problems now be a good time to pause the video
Otherwise we're going to move on and finish up with these last two word problems
So example 4 a path of a football thrown across the field is given by the following equations where x represents the distance in feet
The Ball has traveled horizontally, so that's important to know
so x is the distance in feet
traveled horizontally and
Y represents the height okay, so there's a football
beings thrown across the field so there's the ball
So my x is the distance this way and my y is my height
Okay, so how high it is
In feet above the grunt about how far has the ball traveled
Horizontally when it returns to the ground okay?
Which one will be talking about returning to the ground that's going to be your up and down, so that is your y?
Okay, and what's true about your y if you're on the ground?
Well if I'm on the ground my height is 0 so I'm going to let y equal 0
And what I want to know is how far is a ball traveled horizontally. Well, I'm going to solve it for x I
See that this one is probably going to be best to solve by quadratic formula, or you could graph it
So I'm going to solve it with the quadratic formula
so my a is negative .005 my b is 1
My C is five if I plug it in I have x is equal to negative 1
Plus or minus the square root of B. Squared 1 squared minus 4 times my a
times my c
all over 2 times 8 so 2 times negative 0.005
so x equals negative
1 plus or minus if you type all that into your calculator
Underneath the radical it's going to simplify to 1.1 all over negative 0.01
Okay from here you would get two answers
If you do x is equal to negative 1 plus the square root of 1.1 over negative 0.01 you get?
negative
4.881 okay, we're talking about the distance travels horizontally okay, so we're going this way
We do not want a negative distance. So that one is out
Then I go ahead and I try x is equal to negative one minus square root of 1.1
Divided by Negative 0 . 0 1 if you do that on your calculator you get?
204.881, so this one would be my answer
The distance horizontally is measured in feet so I would label it feet
So basically what they're saying is when the ball hits the ground
over here this distance
was 204. 881 feet
That is example 4 if you have questions, please let me know
Example five an object is thought shot straight upward into the air with an initial speed of 800 feet per second
That's talking about the velocity that 800 and notice it's in your equation right here. Okay, the height h
that the object will
The Height h that the object will be after t seconds is given by this equation
when will the object reach a height of
1000 feet so what they're saying this time, so I have something being shot up
And it's going up and up and up and up and up okay
down here
This distance is my time. They want to know when my height is
10,000 feet so if they want to know when my height is 10,000 feet I'm going to plug in 10,000 for h
So I really have
10,000 is equal to negative16t squared
plus 800 t
This is one right. It's quadratic. I'm going to get everything to one side so 0 is equal to negative 16t squared
plus 800 t
minus ten thousand
Okay, you could try and take out a gCF
and
Factor or you can use the quadratic it's up to you
I'm going to go ahead and use the quadratic so it's one my a is negative 16 my b is
800 and my C is negative ten thousand
So quadratic opposite of b so negative eight hundred plus or minus the square root of B
Squared so 800 squared minus my four times my a times my c
all over 2 times a 2 times negative 16 on
my calculator if I type it in
I'll have x is equal to negative eight hundred plus or minus all of this will simplify to the square root of
Zero all over negative 32 so there's only going to be one solution
So negative eight hundred divided by negative 32 will give me
25 and what is x representing or in this case? It's t it is
Time and the time is in seconds so at 25 seconds
the object shot up so 25 here will be at
10,000 feet
All right, that is the end of unit unit 4 video 7 so if you have any questions, please ask otherwise
This is the last video in unit 4 so have a nice day, and I will talk to you later
