The quadratic formula is a formula to
help you find the roots of quadratic
equations. We already know how to find the roots of some quadratic
equations, but what if we just have a
very generic equation like ax squared
plus BX plus C equals to 0? Let's just
say that these numbers here a B and C
are all constants. So they're not
changing. They're just a single number,
they could be fractions or decimals or
whole numbers. It doesn't matter. They
just need to be constant. Well, in cases
like these, where finding roots are
not immediately obvious, when we can't
factor our equation any further than we
already have, this is when you use the
quadratic formula. The quadratic formula
is, of course, a formula that helps you
find the roots of harder to find roots
in quadratic equations. So the formula
will look like x equals negative b plus
or minus (So you'll end up with two
answers here) plus or minus the square
root of b squared minus 4ac and it's all
over 2a.
These a b and c terms correspond at the
a b and c in our equation up top here.
What we call the standard form of an
equation the standard form has to equal
0.
And it has to be in this format with the
a b and c as constants. Now, one of those
constants may be 0. So you may not have a
bx term or you may not have
a x-squared term or you may not have a C term.
But, you know it's just those numbers
need to be there even if they're zero
even if they're a fraction even if
they're a decimal. So you need to have
this standard for all equal to zero. And
then you need to know this quadratic
formula. Especially since you're going to be
using this a lot in later algebra. So, to
give you an example of how to use this
equation, let's say we have an equation
like 5x squared plus 6x minus 2 equals 0.
When we try to factor this regularly it
doesn't work out so well. We end up with
things that we don't really know how to
factor just in our heads. So we'll have
to use the quadratic equation, sorry, the
quadratic formula. So we'll write it out
here. x equals negative b plus or minus
square root b squared minus 4ac all over
2a. You will probably have to memorize
this equation at some point. There's lots
of places you can go that'll teach you
how to memorize it sometimes through
song,
sometimes through poetry, or what have
you,
but you probably will need to memorize
it at some point like I have. So let's
plug in like a B and C values here. We
have our 5 as our a value our 6 as our B
value and our negative 2 as our C value. When we plug these in, we get x equals
negative 6. So we're just using the
constants in front of the X values. We're
not actually putting the X values into
our quadratic formula.
So negative 6 plus or minus the square
root of 6 squared minus 4 a c and then
it's all over 2a. Now, when you solve this. I'll just
go ahead and do this really quickly.
Since we already know how negative 6
plus or minus the square root of
everything under that square root here.
It's just gonna be 76 here. Then it's all
over 10. Now, you could leave it like this.
Remember that plus or minus means you're
going to have two answers. We're gonna
simplify it just because you know how
and it's always good to get a little bit
of practice. So, if we take our 76 here
we'll get 2 and 38 and then 2 and 19
that's our prime factorization we have a
perfect square of 2 and a 19 left over.
So that would give us a negative 6 plus
or minus 2 times the square root of 19
all over 10 and then, I'm running out of
room here but, we can also write that as
negative 3/5 plus or minus the square
root of 19 over 5. Sorry that's a little
messy, I ran out of room. To give you
another example of how to convert into
standard form, we can take something like
2x plus 2 equals negative 6x minus 1. So
remember our standard form yet we have
to have a x squared plus BX
plus C equals zero. Well, we don't have
our standard form on the left side and
we don't have zero on the right. So let's
go ahead and add six to both sides of
this equation here. Sorry, 6x to both
sides of this equation. So we'll get 2x
squared plus 6x plus two equals negative
one. Okay, well now our left side looks
right but the right side is not equal to
zero and that's a really important to
remember. It needs to equal zero
in order for the quadratic formula to
work. So if we go ahead and add one to
each side we'll get 2x squared plus 6x
plus three equals zero. Now our equation
is in standard form and we can use the
quadratic formula to solve. So plugging
in here we need an x equals negative B.
Our B is our six here so negative B plus
or minus square root B squared minus
four times a times C so two is our a and
three is our C and all over 2a. We simplify, bring
this up here, we'll get a negative six
plus or minus the square root of 12
all over 4. And then, of course, if we
want to, we can find the prime
factorization of our 12. So you'd get and
negative six plus or minus two times the
square root of 3 all over 4 and that
would simplify to a negative 3 over 2
plus or minus root 3 over 2. So you can
see here that these roots that we found
are quite complicated, so we wouldn't be
able to find these just by the factoring
that we already knew how to do. It's
really important that we know the
quadratic formula for any kind of weird
roots that will have radicals or other
kinds of symbols in them that you'll
learn about later. So the quadratic
formula is really really an important
I'll write that down one more time just
so you guys can remember it. The
quadratic formula is x equals negative b
plus or minus square root b squared
minus 4ac and that's all over 2a. And that's
our quadratic formula.
