If you're wanting to learn how to
memorise the quadratic formula,
you've come to the right place!
G'day.
Welcome to Crystal Clear Mathematics
where it is easier than you think.
I'm your host, Graeme Henderson.
There are many, many ways
to learn mathematical formulae,
and I've created links to some quite good
ones
in the description below this video.
The method I want to show to you here
is slightly different.
The reason I like it, and the reason
I adopted it and used it when I was in school
is that I wanted every formula I learned
to be linked to the mathematics I already
knew.
And the best way I knew to do that
was to learn the derivation.
And, surprisingly, learning the derivation
does not take long and it's not difficult,
but it has tremendous benefits.
So, what I propose to do is
to write up the general quadratic equation
and derive the formula from it.
When I've finished doing that I'll explain
how to memorise it and learn it well.
So, let's get started.
The quadratic equation is y = ax^2 + bx +
c.
These kinds of equations occur
right throughout mathematics.
You can find them cropping up in geometry,
trigonometry, calculus ...
and the list would go on and on.
So it is quite important
to learn this skill ...
how to solve them and how to use them.
Of course, in this form,
if we graph them,
we get a kind of curve
known as a parabola.
If you choose any particular x-value
and substitute it in the equation,
you'll get a value for 'y' which will
tell you some point on the parabola.
Substitute another x-value,
you'll get another y-value.
Now, when graphing parabolas,
one of the key bits of information
we'd like to know (or one of the key points,
or pairs of points, that we want to know)
are the points where the parabola
crosses the x-axis.
We find these because
the equation for the x-axis is y = 0.
Remember, it's the x-value
that changes along the x-axis
(which is why it's called the x-axis)
but the y-value is zero because
you've not gone up or down the y-axis at all.
And, if we set y = 0, so zero equals this
or this equals zero, and solve it,
we actually find these two points
(known as the roots).
And that is what
the quadratic equation is for.
It's for solving not this equation
but THIS equation.
So I'll conveniently get rid of the 'y ='
and we're going to start with this
and derive the quadratic formula
which, in essence,
is a rearrangement of this same equation.
Now, the method we use for solving
the quadratic equation
is called 'completing the square'
so, in the very act (or process)
of deriving the formula,
you're actually practicing another skill
that you need to understand.
Now, to complete a square
is a bit more difficult
when you don't actually have numbers
as coefficients.
So, we've got to practice
our algebraic skills quite a lot.
We don't like, as a rule,
to try to complete a square
when the leading coefficient
is not one.
So, we want to remove this 'a'
by dividing everything by 'a.'
These, of course, divide out.
We have one lot of x^2
plus b/a lots of x plus c/a
equals zero divided by 'a' is zero,
conveniently.
This is a big improvement,
and that's your first step.
Your second step is to remove this constant
because 'c' and 'a' are both constant terms.
We need to remove that because,
to complete this square,
we're going to put a different value there.
So, we subtract it from
both sides of the equation
and get (I'm going to write it over here)
... so, we've moved it over there.
We now need to find a value
that will complete this square.
And when the leading coefficient is one,
the rule is quite simple.
We halve the coefficient of 'x' ...
and the easy way to do that
is just double the denominator
(as though we multiplied the fraction
by half)
and square it ...
so it's actually (b/2a)^2.
That completes the square.
So, it's half of that
(by putting a two in)
and squaring it.
Now, of course,
we have an equation here,
so we can't add this without adding it
to the other side as well.
Now the equation is still balanced.
That step, from here to here,
is completing the square.
Now, because this square has been completed,
it is a perfect square, and it is (x + b/2a)^2.
Now this side we can simplify a bit.
We'd like to try to get this fraction and
this
Fraction combined into the one big fraction.
Let's expand this first,
because it is a square.
So, b^2 over ...
(and the square applies to the 2
and the 'a' as well)
... so 2^2 = 4 and a^2 is a^2 of course
... minus c/a.
And, to combine these,
we want the denominators the same.
'a' already divides into 4a^2.
It divides 4a times ...
that is, a times 4a makes 4a^2.
So, that's exactly what we do.
We multiply the top and the bottom
of the fraction by 4a.
So we have 4a^2 here and 4a^2 there
which means our resulting fraction
will have 4a^2 on the bottom.
This part has a b^2 on the top
... minus ...
and this part has 4a times c.
And you can see here
part of the quadratic formula appearing.
I'll just copy this again.
Now we take the square root of both sides
to get rid of the square.
That gives us x + b/2a =
the square root of this.
Now, the square root of 4a^2 is 2a.
And, the square root of this
we simply can't do
so we put a square root sign or a
radical sign over it ... b^2 � 4ac.
Now we have a slight problem.
Whenever you take the square root
of both sides of an equation,
where pronumerals are involved,
you should use absolute value signs ...
which means we have to allow
for a positive or a negative result.
To isolate 'x,' we must remove this
last term and take it over there.
Fortunately, it's still over 2a
so we can make it all over 2a.
This +b, over the other side, becomes
a -b ... +/- square root b^2 � 4ac.
And there is our quadratic formula.
Practising this derivation is
an extraordinarily useful tool.
How long does it take?
Once you know what you're doing it takes
maybe two minutes � three minutes.
Do that once every afternoon
following this simple pattern:
Set time aside without distractions
... that is, you must focus.
If you focus
you're going to benefit from this.
If you try to do it while you're
watching TV or listening to music or something,
it's far less likely that it'll 'stick.'
So, just for these two or three minutes,
get rid of all distractions, concentrate,
and derive the formula.
That's it!
When you get to the end,
just check with your master sheet
that you have, in fact,
followed all the steps properly.
You only have to do that once every afternoon
for two, three, four, five, six afternoons
...
however long it takes ...
until one day, you're doing it
and you think,
"I have so got this! This is so simple!"
That means you don't have to do it any more.
So, I'll just put here ...
"Once per day."
Now, you can do it more often if you wish.
The third principle (there are four) ...
the third principle to learn this well
is that
(at the same time you're doing this)
is to use the formula
to solve equations.
That means, get your text book,
find the exercise
where you're given a whole lot
of these quadratic equations to solve,
and do a handful of them.
Now, the more of these you do, more intensively,
the better you're going to learn.
This is what we call massed practice.
And it's vital for good long-term
recollection of the skill and the facts.
And the last thing is to do
what's called 'distributed practice.'
And that is, to get your diary and
to ask yourself three fundamental questions:
"Am I fast enough?"
Normally, that's the second question I ask.
The other question is, "Do I understand?"
These are the two questions
you should be asking yourself
all the time
you're doing the massed practice.
When you can tick them both and say,
"Yes, I understand how to
derive the formula,
I understand how to use the formula
and I'm fast enough at doing them both,"
then you ask the third question which is,
"How long before I forget?"
Because, forget we do!
It is very simple (easy) to forget.
Now, if you're honest with yourself
(this is a very personal question) ...
you might feel that you can go
six months without forgetting.
Some students will.
You might decide
it might only be two weeks.
But, somewhere in between,
there'll be a time when you think,
"Yes, I think I'll understand it here,
but not here."
That's when you write
a little note in your diary
to revise this derivation
and then to use the formula again
and to ask yourself
the three questions again.
If you choose this date properly,
the whole practice of this ...
the whole study of it ...
might only take you ten minutes.
If you time it badly,
you will have to relearn the derivation
and possibly [have to] relearn
the use of the formula.
So, it pays to be very honest with yourself
and to err on the side of caution.
So, if you think you'll be right
for about two months
but you're not sure about three or four,
put a note in your diary for two months' time.
I know I've repeated myself a little ...
that's part of the way I teach ...
but I hope this helps:
Do the derivation; do the practice;
make sure you do it
without interruption
(that you actually set time aside
to do it);
ask the questions;
and use your diary to make sure
that you recall it at intervals of time
right up until your exams.
Thank you for watching.
