Welcome to solving quadratic functions by quadratic formula and all methods the best way to use this video is to take notes while I'm teaching
Pausing whenever necessary and every time the video arrives at AU try
Pause and attempt the problem the solutions will be shown when you press play along with the work
We'll cover the following topics in this section
So what does it mean to solve a quadratic function? Well, it means we're looking for the
X-intercepts, we don't usually call them x-intercepts. A lot of times. We call them roots zeros or solutions
There are three methods for solving quadratics graphing factoring and quadratic formula when you solve it algebraically
Three things can happen you could end up with two zeros one zero or no real zeros
what that looks like on a graph is two zeros two places it hits the x-axis one zero one place it hits the
X-axis and no zeros where it's not hitting the x-axis at all
so with the quadratic formula
This is the method that works for all
Quadratic functions. So if we were asked to solve this
I always try and factor it so when I try to factor it I realize oh wait
I can't divide all my coefficients by three so I say all right. I guess we'll try graphing it
But when I try graphing it I get these numbers that don't exactly work out
I need another method and that's where the quadratic formula comes into play. It works for every single quadratic function
So the way the quadratic formula works is it's the opposite of B plus
Or minus the square root of so it's a big square root symbol
b squared minus
4ac
all over big fraction bar 2a
So these are the all the methods of solving quadratics plus my opinion
You can't solve quadratics by graphing B. We rarely do that because it just takes so long
Factoring if we can because usually that's the quickest and quadratic formula if we can't factor
So this first example solve the quadratic function?
So we can't factor it divide by two divide by two and I don't want to graph it
So we're going to use the quadratic formula and label each one a b c
x equal the opposite of B plus or minus the square root of
B squared use a parentheses minus 4a
see
Extend if you need to all
Over two times a now take a moment and double-check that your radicals big enough and that your fraction bar is big enough now
The next thing we're gonna do is type what I've highlighted
Only oops forgot a parenthesis there what I'm highlighting into the fraction to the calculator
Do not type the radical do not type that
So we're gonna get x equals 7 plus or minus and then radical all over 4
So go ahead and type that into your calculator. You should get 73 if
You did not get 73
It's because you typed in the square root if you've got the square if you got the value
8.54 it's because you type that radical symbol in if you've got
Let's see here
If you got negative 25, it's because you forgot to type in your parentheses
And the reason why we are done is we cannot simplify square root of 73
There are no perfect squares that go into it. This is two solutions. They're irrational seven plus radical
73 over four and seven minus radical
73 over four
All right. Let's take a look at example two
Solve the quadratic function. Let's see if we can factor it. We have a coefficient of a 1 here
Is there any number what multiplies to 5 does it add to 5 so we've got 1 times 5
No way to add it. So we have to use the quadratic formula a b c
So x equals the opposite of b plus or minus big square root symbol
B squared minus 4 ay
C all over
2a
Now again type underneath the radical only don't actually type that radical symbol
So we get x equals negative 5 plus or minus the square root of all over 2
So what happens you type that into your calculator you get five?
We cannot simplify the square root of five at all
So we get negative five plus square root of 5 over 2 and negative 5 minus square root of 5 over 2
We're gonna do a few more examples together
Solve the following quadratic function using the quadratic formula. Maybe we can factor Oh
They say we have to use the quadratic formula
Alright, so there we go
So one a B C X
Equals the opposite of B plus or minus the square root of B squared minus 4
ay C notice how I have to extend my square root all over giant fraction to a
Remember type just underneath the radical not the actual symbol itself
negative 3 plus or minus the square root of when you type that in your calculator you
Get 49 well 49 is a perfect square so
This is equal to negative 3 plus or minus 7 over 2
And we'll split that up. We get negative 3 plus 7 over 2 and negative 3 minus 7 over 2
so that gives us 4 over 2 is 2 and negative 10 over 2 is negative 5
These are our zeros
Well, let's just do a quick check. Could we have factored this what multiplies to negative 10 but adds to 3
positive 5 negative 2 negative
5 positive 2
Alright we'll do a couple more together
Solve the following quadratic function. Alright, it's 5 we can't put 5 into these so we've got to use the quadratic
formula so x equals the opposite of b plus or minus the square root of
b squared minus 4ac
All over 2a
And you're going to type just underneath the radical here not the actual radical itself negative 3 plus or minus the square root of all
Over 10 and you get negative of 11:00
This is not a real number so the solution here is no real zeros
But sometimes you're gonna want to simplify it even more. Maybe they want to know the imaginary number if that's the case
We know how to simplify this it would be negative 3 plus or minus. I
Square root 11 over 10, if you're not sure how I got that I go back to the simplifying radicals video and imaginary numbers
Let's take a look at what the graph would look like for no real zeros
well
we know that a is positive so it's gonna look like this the graph is probably gonna look something like that and
When I graph it in desmos, that's exactly what I see
all
Right last one together solve the following quadratic
Well, I cannot factor because they don't all divide by 4. So we'll write ABC and go ahead and use the quadratic formula
Opposite of B plus or minus the square root of b squared minus
4ac
all
over
2a
x equals remember to type just underneath the radical into your calculator a
Lot of times the mistakes that happen have nothing to do with the math and all to do with the calculator
So be really careful typing that in
We will get positive
289 and
if you type that in your calculator 289 is a perfect square it is equal to
17 and so we can write this in two different ways. We get negative seven plus 17 over eight and
Negative seven minus 17 over eight
Get 10 over eight, which is 5 over 4
We get negative 24 over 8, which is negative three. These are our solutions
So go ahead and give these problems here
a
try and
Then press play when you're ready to see the answers
Now let's take a look at solving quadratics using all methods
At this point, you know how to graph
You know how to factor and you know how to use the quadratic formula
So now I'm going to give you these four problems
I'm gonna say you can solve them using any method you'd like and explain why you chose each method
When I when you press play again, you will have all my work plus my reasoning
Maybe you decided to use quadratic formula for all of them. That's perfectly fine as well
And that concludes solving quadratic functions by quadratic formula and all methods
