Hey ladies and gentlemen, this is Mrs. Beckmann here. Today we're going to be going over unit 4 video 7.
So today we're going to be talking about the quadratic formula.
Now this is something that you did in algebra,
So it's going to be a little bit of a review and then we're going to be taking this to the next step.
So let's take a look at this first problem.
So it says competitors in the 10-meter platform
dieting competition jump upward and outward before diving into the pool below.
The height of a diver and meters above the pool after T
seconds can be approximated by the equation: H is equal to -4.9t squared plus
3t plus 10. How many seconds did it take for the diver to hit the pool?
So one of the first methods we learned was to solve by factoring, but since our a value is going to be a
decimal, it is going to be really hard to factor because it's hard to find numbers that
multiply to a decimal.
And so that's not going to be our best method. Can we solve this by graphing?
Yeah, we can, let's just take a look at that graph here.
So I have this all typed in and so to solve it by graphing,
I want to look at the two spots that it hits zero. So it hits at -.155 and 1.767
Now if we're talking about time T is representing time, or in this case X, we can't have a negative time.
Okay, we have an invented time travel yet.
So this 1.767 is going to be the way that we find the zero.
So that's going to be the number of seconds. Now a lot of us don't have a graphing calculator.
And so this is going to be yet another way that you can solve quadratics that we are going to be learning today.
Alright. So taking a look at this, you have the quadratic formula.
So whenever you're given something that's ax squared plus bx plus c is equal to zero. That's really important, you do need to make sure that it is equal to zero
Okay, and that your a is not equal to zero because if your a is equal to zero
then you don't have an x squared term. So it's going to be given by a specific formula.
Alright.
So that's going to be x is equal to opposite of b plus or minus the square root of b squared minus 4ac all over
2a. Frankly you're going to use this formula so much that most of the time by the time the students take quiz
they already have it memorized.
So let's go ahead and let's take a look at this first one.
So if a discriminant is greater
than zero, so the discriminant that's going to be what's underneath the square root, that b squared minus 4ac.
So if it's greater than zero, so basically if you calculate it out and it's positive
that means you're going to have two real solutions.
So an example of that would be the square root of 50. Now if what's underneath the discriminant is equal
to zero, so you get like basically the square root of zero, then you're only going to have one real solution.
Okay, then if the discriminant is less than zero, which basically means that it's negative.
So if it's negative underneath then there's no real solutions. Okay?
Now in our previous lesson we learned what we can do if we have a negative
underneath the square root, and that's to take out the letter i.  Ao there might not be any real solutions,
but there will be two complex solutions.
So that's what we're going to be adding to what we've already know about the quadratic formula today.
We're going to be learning how to deal with complex
solutions as well as our real solutions. So let's go ahead and let's take a look at this.
So let's take a look at this first one.
Okay. So for this first one we have x squared minus 12x is equal to 28.
So being able to use the quadratic formula, it has to be equal to zero, so I'm going to start by subtracting
28 from both sides. So then I get that x squared minus 12x
minus 28 is equal to zero, okay.
So now I want to go ahead and I want to do the quadratic formula. So I'm going to do the quadratic formula.
So I have my a. My a is always with the squared term,
so since that's just an x squared my a is 1.
My b is always with the x term, so my b is negative 12 and my c is my number that's alone, negative 28.
So now that I have all of those figured out,
what I'm going to do next is I am going to try and plug in what I do know, okay.
So I'm gonna plug this into the formula. So I'm going to have x is equal to opposite of b.
So instead of writing in negative 12, I'm going to write a positive 12 plus or minus the square root of b squared minus
4 times my a value, which is 1,
times my c value,
which is negative 28, and all of that's going to be divided by 2 times 1.
So what I like to do next is figure out what's underneath the square root. So in my calculator I do
parenthesis negative 12, parenthesis, squared minus 4 parenthesis one parenthesis parenthesis negative 28.
So I type all of that in at once and
when I do that underneath I end up getting
256, and that's going to be divided by 2 times 1.
Well the square root of 256,
256 is a perfect square. So I end up getting 16.
So then I get 12 plus or minus 16 divided by 2.
So now I'm going to split this into two versions.
So I have 12 plus 16 divided by 2 and 12 minus 16 divided by 2.
Now when I do this in my calculator a lot of students like to type this all in at once and that ends up screwing
them up. You do need to add the top before you can divide.
So 12 plus 16 is 28 and then I'm going to divide that by 2.
Well 28 divided by 2 is 14.
So that's going to be one of my answers, 12 minus 16 is negative 4.
Negative 4 divided by 2 is negative 2, so 14 and negative 2 are going to be my answers for number one.
Now, let's take a look at number two.
So for number two
I want to start by identifying what my a, my b and my c are.
So my a is always with the squared term,
so here my a is 2. My b is always with the x term, so my b is 4. My c is my
constant, so that's going to be negative 5. So once I know what my a, my b and my c are,
I'm going to plug that in, so I have x is equal to opposite of b plus or minus the square root of
b squared minus
4
times a
times c all over 2 times a.
 
Okay, so now I need to simplify that out. So I'm going to plug in what's underneath the square root into my calculator.
So if you're struggling to type this into your calculator correctly,
please let me know so I can show you. I might suggest that you calculate along with the video.
So you make sure you're doing it correctly.
So then 2 squared minus 4 times 2 times negative 5, that's going to give me
56 underneath the square root. Now that's going to be divided by 4.
So this time instead of writing out a decimal,
I want to see if I can figure out what this is as a simplified square root.
So I'm going to start by trying to break down the square root of 56. Now 56 is not a perfect square.
But I am going to see if I have any pairs that I can take out.
So, 56 I can split up into 4 and 14.
4 I can break down to 2 and 2 and 14 I can break down to 2 and 7.
So here I can take out a pair of 2s.
Okay, so I'm gonna write that, so my negative 4 plus or minus that's going to stay the same, but instead of writing the square
root of 56, I want to take out my pair of 2s on the outside.
So that is going to become 2 and then left underneath is 2 times 7, which is 14.
So I have 14 that's left underneath the square root and that's all going to be divided by 4.
So what I notice next is these numbers that are on the outside of the square root are all divisible by 2.
And I cannot do any dividing or simplifying unless every single piece that's on the outside is divisible by 2.
So I'm gonna go ahead and divide each of those by 2.
So negative 4 divided by 2 is negative 2.
Plus or minus 2, square root of 14 divided by 2 is just going to give me the square root of 14.
4 divided by 2 is 2.
So this right here is your completely simplified version.
So for the purpose of algebra 2 I do want you writing these as simplified radicals.
So that means if it asks you to solve it using the quadratic formula,
you cannot just graph it and write the decimal you do need to do the quadratic
formula so that you can give it to me as a simplified radical.
So let's take a look at number three.
So let's start by deciding what our a, our b and our c are. So my a is always with the squared term.
So my a is 3. My b is with the x term, so my b is five and my c is
what is just a number. So now I need to plug this into the formula so I get x is equal to
opposite of b plus or minus the square root of
b squared minus
four times a times c all
over 2 times a.
The next thing I'm going to do is do what's underneath the square root.
So I get negative 5 plus or minus the square root.
So then I have 5 squared minus 4 times 3 times 4 and when I type that in my calculator I get a negative
23 and that's going to be divided by 2 times 3, which is 6. So here I have a negative 23.
So in order for me to be able to do this
I'm going to have to take an i out because of that negative square root.
So I have negative 5 plus or minus, then I'm gonna take i
out of the square root.
And when I do that then I'm just left with 23 underneath the square root.
And then I'm going to take that and I'm going to divide that by 6.
So here I get negative 5 plus or minusi times the square root of 23 divided by 6.
Well 5i and 6 have nothing that can divide them so that can't be simplified and
23 is a prime number so that cannot be simplified either.
So I want to take a look at what this is going to look like as a graph. So I have
3x squared
plus 5x
and then plus 4. Okay. So what you're going to notice about this graph is at no point
does it cross the x-axis.
So now let's take a look at these final two word problems.
So number 4 says the path of a football thrown across a field is given by the equation y is equal to
-0.005x squared plus x plus 5, where x represents the distance in feet the ball travels
horizontally and y represents the height in feet of the ball above the ground. About how far has the ball traveled
horizontally when it returns to the ground? Well the fact that it's asking how far does the ball travel horizontally
means that we need to find x and then it's asking when the ball returns to the ground.
So when you're on the ground your height is zero.
So that means that my y right here is going to be equal to zero.
So I have zero is equal to -.005x squared
plus x plus five. So in order to solve this I am going to use the quadratic formula.
So I need to start by figuring out what my a, my b and my c are. So my a is always with the squared term,
so that's  -.005. My b is always with my x term, well since it's just an x,
that means my b is 1, and my c is always my constant at the end.
So now let's plug this in. So x is equal to opposite of b plus or minus the square root of
b squared minus
four times a which is -.005 times c all over two times
my a value.
So my first step is to get what's underneath the square root and simplify that out first. So I do one squared minus four
times negative five times five, and when I do that
I get 1.1, and that's going to be divided by 2 times -.005.
Which going to give me a -.01.
So now I need to split this into two versions.
So when I do that,
I'm going to have -1 plus the square root of 1.1 divided by a -.01.
I'm also going to do -1 minus the square root of 1.1 divided by a -.01.
Okay, so to plug this in your calculator, you always want to simplify out the top first.
So I would do a -1 plus the square root of 1.1.
Hit enter, see what that equals and then divide by a -.01.
So when I do that for the top one, I get -4.881.
So that's not gonna work for that one.
Then for this next one -1 minus the square root of 1.1 hit enter see what that equals and then divide by
a -.01, and when I do that I get 204.881 and
then I want to see what my x represents. It says the distance in feet.
So that means that I need to label this in feet. The answer for the top one doesn't make sense because we're not traveling backwards.
We're looking at when we hit the ground
traveling forwards which means 204.881 is going to be our answer for number four.
Now, let's take a look at number five.
So it says an object strap shot straight upward in the air with an initial speed of 800 feet per second. The height hbof
the object will return in t seconds is given by the equation
negative 16t squared plus 800t. So notice that the 800 that they give us is included in the problem.
So a lot of students try to put it in the problem again,
but it's already included in there and it wants to know when the object is going to reach a height of 10,000 feet.
So that means that my h is going to be 10,000, so I am going to have
10,000 and that's going to be equal to negative
16t squared plus
800t. So in order for me to do the quadratic formula,
I need to have this set equal to zero. So I am going to subtract
10,000 over to the other side.
So when I do that I get a negative 16t squared plus
800t
minus 10,000 is equal to zero. So now I'm ready to do the quadratic formula on this problem.
So my a that's what's going be with the t squared. So that's going to be negative 16. The b is with the t,
so that's going to be 800, and the c is going to be that negative 10,000.
So now let's go ahead and let's do the quadratic formula. So t is equal to
opposite of b plus or minus the square root of
b squared minus
4 times a times
c
all over 2 times a.
So then I want to take what's underneath the square root and I want to simplify that first. So in my calculator I do
800 squared minus 4 times negative 16 negative 10,000, and when I do that I end up getting
0, and then 2 times negative 16 is negative 32.
Now remember at the beginning of this lesson we talked about if you get 0 underneath the square root
that means you're only going to have one solution.
So this is only going to have one real solution.
The square root of 0 is just 0, so that means my answer is simply going to be negative 800 divided by negative
32. So negative 800 divided by negative 32 is going to give me a positive
25, and t is representing time in seconds.
So that means that my label here is going to be seconds.
So 25 seconds is my answer for number 5. So that concludes your note video for today.
Please let me know if you have any questions. Thanks for listening.
