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.