in this video we'll talk about how to
solve quadratic equations using the
quadratic formula. So, remember a
quadratic equation is one of this type.
A, B, and C are constants. So. here's a
little example 2x squared plus 3x minus
7 equals 0 that is a quadratic equation.
Now, we've seen we can solve equations of
this type via factoring, but sometimes
you have something where it's just like
a big pain to factor or you try to
factor and you get totally stuck. So, if
that's the case then you can always try
using this thing called the quadratic
formula. So here's the quadratic formula.
The quadratic formula basically says
that if you have a quadratic equation in
this form then the solutions to it are
actually given by this formula here
where A, B, and C are the coefficients of
the x squared term, the x term, and then
like the constant term in the quadratic
equation that you start with. So we're
gonna look at an example. I'm going to
show you guys how you use this formula.
So we'll find all solutions to 4x
squared plus 3x minus 6 equals 0 and
we're gonna round our answers to 2
decimal places.
Ok, so our equation is 4 4 x squared plus
3x minus 6 equals 0. Now in order to use
the quadratic formula to find the
solutions here we need to first identify
what a B and C are so I'm gonna first
identify a B and C. In other words I need
to look at this and determine what is A,
what is B, and what is C. So here we have
A is 4 B is 3 and C is negative 6. Now
the equation I'm looking at here, it's
already in like the correct order, in
other words I already have it written in
it as x squared plus BX plus C equals 0.
if your equation was not
this order you would have to kind of
like rearrange it using algebra until it
had the x squared term first, then the x
term, and then the constant term and then
equals zero. If it wasn't like that you
would rearrange it.
Okay, now the quadratic formula says that
our solutions are going to be given by
this equation x is equal to opposite of
B plus or minus the square root of B
squared minus four times a times C all
divided by 2a. And so what I can do is, to
find the solutions, I can actually stick
into this equation, I can stick B, A, and C
in each of these spots here. So I'm gonna
replace B with three, it's my right three
in the B spot, and I'm gonna do that
everywhere I see a B, then what I'm gonna
do is instead of A,  I'm going to write
four, so I'm gonna write minus four times
four, and then instead of C I'm gonna
write negative six so I'm going to put
times negative six in here and then I'm
gonna divide the entire thing by two
times A, which we have determined is four.
And this is gonna give us some solutions
to the equation. So, the most common
mistake I see people making here is,
there's two actually, one of them is just
accidentally replacing one of these
letters with the wrong thing. So, like in
the C spot I need to make sure I put
negative six and not something else, like
four or three, and then the other mistake
I see people make is not doing this
division line all the way. So, it's like
every single thing on the top, like the
whole entire thing, has to be divided by
the thing on the bottom here. So extend
this line all the way. Okay, from here we
need to simplify. So, I always start by
simplifying inside the radical, so I'm
going to write plus or minus, I'm gonna
simplify everything in the radical. So,
I'm gonna do 3 times 3 that's 3 squared
so if I do that I get 9 and then I'm
gonna do 4
times four times negative six so that
gives us minus four times four times
negative 6 is minus 96. So, you can use a
calculator for that if you want and then
2 times 4 in the denominator that's
actually just 8. Okay, now I have minus a
negative here, so minus a negative and
double negative turns into a positive, so
this becomes square root of 9 plus 96
and then I can do 9 plus 96 to get 105.
Okay now where do I go from here? I
actually have two answers here:
so the plus or minus means that we have
like two different answers ,one of the
answers is x is equal to negative 3 plus
square root of 1 over 5 105 over 8 and
then the other answer is x equals -3
minus square root of 1 over 5 plus 8. So
if you guys were wondering what the plus
or minus means it essentially means we
have two answers, one of them where
there's a plus here and then another one
where there's a minus here. So you have
two answers now we can leave this as is
which would be called an exact answer and if you can simplify the radical
you can go ahead and try to do that or
we can type this into a calculator to
find a rounded answer. So if you take
your calculator out and type in negative
3 plus square root of 1 over 5/8 you
would want to be careful and put a
parentheses around this whole thing so I
would type into my calculator
parentheses negative 3 plus square root
of 105 or 5 parenthesis closed and then
divide by 8. And if you don't put the
parentheses around the top stuff, it's
only gonna divide the 1 over squared of
1 over 5/8, but it's not gonna do the
negative 3 over 8, which is important. So
put this into your calculator and round
to two decimal places and you should end
up getting the following 0.91 for the
first one or x is equal to negative 1.66.
So that's rounded to two decimal places
I'm putting this squiggly equal sign and
that means, like that I've rounded or
it's an approximate answer. So these are
our two answers rounded to two decimal
places.
