Hi. In this video, we're going to talk about
the quadratic formula and how to use it with
an example. So if we have a quadratic equation:
a x squared plus b x plus c equals zero, this
is called standard form. a b and c are just
numbers. The quadratic formula allows us to
solve any quadratic equation, even those that
cannot be factored. The quadratic formula
is: x equals (put a big fraction). On the
top is negative b plus and minus the square
root of b squared minus 4 a c, all of which
is divided by 2 a. That might seem like a
lot, but we have a song to help us memorize
it.
It goes like: x equals negative b plus and
minus the square root of b squared minus 4
a c, all over 2 a.
I know I'm the world's worst singer, but I'm
not afraid to sing the quadratic formula song.
x equals negative b plus and minus the square
root of b squared minus 4 a c, all over 2
a.
When I teach this lesson to my classes in-person,
I sing it about twenty times until everyone
in the class is singing along. My students
usually get kind of frustrated with me, but
they remember it! So, I would strongly suggest
you sing this along a few times. I'm not going
to sing it twenty times here because you're
just going to turn off the video. But, let's
do it one more time. Try to sing it with me!
x equals negative b plus and minus the square
root of b squared minus 4 a c, all over 2
a. (That's going to be stuck in your head,
but you're going to remember it, and that's
a good thing!)
So, let's do an example of using the quadratic
formula to actually solve a problem. Here,
our problem is 3 x squared minus 4 x plus
5 equals zero. You could try to factor that,
but it won't work. You could try your whole
life to factor it, and you wouldn't be able
to. However, the quadratic formula allows
us to solve it quite easily. To start, let's
write down what your a, your b, and your c
are. Your a is the number in front of x squared,
so a equals 3. Your b is the number in front
of x, so b equals negative 4. And your c is
the constant so c equals 5. Quick comment
before we go too far: This does need to be
equal to zero if we're going to use the quadratic
formula. If it was not equal to zero, we would
have to do some work first to make it equal
to zero.
But, now that we have our values for a, b,
and c, we can rewrite, or put these numbers
into the quadratic formula. We have x equals
your big fraction. Negative b. Be careful
since b is negative, you do need this double
negative. Lots of people forget that double
negative. Plus and minus the square root.
Your b again, squared. Again, you do need
these parentheses. If you don't put those
parentheses, and you use your calculator,
the calculator will give you the wrong answer.
Minus 4 a c. Your a in this problem is 3.
And your c is 5. All of that is divided by
2 times your a.
Now, it's just a matter of simplifying. We
have x equals a negative, negative 4 (two
negatives makes a positive). Plus and minus
the square root of--you could type all of
this stuff inside the radical into your calculator
if you want to, or we can do it by hand--negative
4 when you square it is a positive 16. Four
times three times five is minus sixty. Your
denominator two times three equals 6.
So, we have x equals four plus and minus the
square root of: 16 minus 60 equals negative
44. All of which is divided by 6. We're not
finished because we need to simplify this
square root of negative 44. To do that, let's
make a factor tree. 44 equals two times twenty-two.
Twenty-two equals 2 times eleven. Your pair
of 2's moves outside of the radical. Since
we had a negative inside your square root,
we have to take an i out of the square root
as well. And the eleven is the only number
left inside the square root. All of which
is divided by 6.
We are not quite finished! Because we can
simplify this! What number goes into 4, 2,
and 6? Well, 2 goes into all three of those
numbers, so we need to divide 4 by 2, 2 divided
by 2, and take 6 divided by 2. When we've
divided all three of those numbers by 2, we'll
have 2 plus and minus i square root of 11,
over 6 as your final answer. Be careful: lots
of people want to reduce this 2 over 6, but
this 2 over 6 does not reduce because this
is really a 1 here in front of the i. The
only time you can reduce is when all three of those
numbers reduce by the same number like we
did here.
So I hope you learned something and that you
have the quadratic formula stuck in your head
now. Let's sing it one last time for good
measure. x equals negative b, plus and minus
the square root of b squared minus 4 a c,
all over 2 a. Thanks for watching. Hope you
learned something!
