Find a formula for solving a quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥
plus 𝑐 is equal to zero. So here’s our equation we’re trying to solve in terms of 𝑥, and 𝑎,
𝑏, and 𝑐 are just constant numbers.
So first of all, we’re gonna divide both sides of the
equation by the constant 𝑎. So the 𝑎’s here cancel out and we’re left with 𝑏𝑥 over 𝑎 plus 𝑐
over 𝑎 is equal to zero over 𝑎. Well that’s just zero. So with just a little bit of tidying up,
this is the equation we’re now trying to solve.
And now I’m gonna subtract this 𝑐 over 𝑎 from
both sides of my equation, so subtracting from the left-hand side and subtracting from the
right-hand side. So 𝑐 over 𝑎 take away 𝑐 over 𝑎 is nothing, and on the right-hand side at zero
take away 𝑐 over 𝑎, I’ve just got negative 𝑐 over 𝑎.
Now I’m gonna use the method of completing
the square on the left-hand side to try and find another form of that left-hand side, which is
gonna help us to solve our equation. So this is our first guess: what we’re going to square in
order to get an equivalent expression to this one.
Now the 𝑥 squared term tells us we’re gonna
use 𝑥 here and then we take half of this coefficient for this term here. So let’s multiply out
these brackets. 𝑥 times 𝑥 is 𝑥 squared, 𝑥 lots of 𝑏 over two 𝑎 is 𝑏 over two 𝑎𝑥, 𝑏 over two 𝑎
lots of 𝑥 is another 𝑏 over two 𝑎𝑥, and finally 𝑏 over two 𝑎 times 𝑏 over two 𝑎 is positive 𝑏
squared over four 𝑎 squared.
Then adding 𝑏 over two 𝑎𝑥 plus 𝑏 over two 𝑎𝑥 gives us 𝑏 over 𝑎𝑥.
Now the first part here 𝑥 squared plus 𝑏 over 𝑎𝑥 is exactly what we were looking for up here,
𝑥 squared over 𝑏𝑎𝑥. The problem is, we’ve got this extra bit on the end, the extra 𝑏 squared
over four 𝑎 squared.
So this expression here isn’t the same as this expression here. We’ve got
an extra 𝑏 squared over four 𝑎 squared. So if I took that away from this term, then I would end
up with what I wanted at the very beginning.
So our first guess was 𝑥 plus 𝑏 over two 𝑎 all
squared would be equivalent to 𝑥 squared plus 𝑏 over 𝑎𝑥, but as we’ve seen down here, that’s
not quite right. It gives us a little bit too much, so I need to subtract that from this in
order to get what I was actually looking for. So hopefully you can see that these two lines are
the same thing; 𝑥 plus 𝑏 over two 𝑎 all squared gives us this expression down here.
subtract 𝑏 squared from four 𝑎 squared, that’s like getting rid of that bit, which just leaves
this bit here, which is in fact exactly what we were looking for. Right so now I’m gonna try and
isolate the 𝑥 plus 𝑏 over two 𝑎 all squared term by adding 𝑏 squared over four 𝑎 squared to
both sides of my equation.
Now we can see that, on the left-hand side, I’ve got negative 𝑏
squared over four 𝑎 squared, and then I’m adding 𝑏 squared over four 𝑎 squared, so those two
terms are gonna cancel out. And now I’m gonna try and manipulate the right-hand side so that,
instead of two separate terms, I’ve just got one big term.
And the way I can do this, I’ve got a
denominator on this term of four 𝑎 squared. I’ve got a denominator on this term of just 𝑎. So if
I multiply the top by four 𝑎 and the bottom by four 𝑎, now my denominator on this term is gonna
be four 𝑎 squared just like this term.
Now by multiplying by four 𝑎 over four 𝑎, that basically
is one, four 𝑎 divided by four 𝑎 is one, so I haven’t changed the magnitude of the number; I’ve
just changed the format of the expression. So when I rewrite that, you can see that I’ve got two
terms with a common denominator, so I can just put that altogether in one big term.
we have it. So rather than write minus four 𝑎𝑐 plus 𝑏 squared, I’ve put the positive term first, so
𝑏 squared minus four 𝑎𝑐 all over four 𝑎 squared. Now I can take square roots of both sides
so that I can see on the left-hand side I’m just gonna find out what 𝑥 plus 𝑏 over two 𝑎 is.
clearly there are two possible answers for the right-hand side. We could have the positive
version of the square root of 𝑏 squared minus four 𝑎𝑐 all over four 𝑎 squared or we could
have the negative version of that. And now what I’m gonna do as well is I’m just gonna
rearrange that second term slightly because the square root of four 𝑎 squared is clearly two 𝑎.
And now I can simply subtract 𝑏 over two 𝑎 from both sides, so that I can leave 𝑥 on its own
over here on the left. So we’re going to subtract 𝑏 over two 𝑎 from both sides of the equation.
So over here on the left-hand side, I’ve got 𝑏 over two 𝑎, take away 𝑏 over two 𝑎, so that leaves
nothing, so those two things cancel out.
And now I’m gonna write this in a slightly different
order, so I’m gonna write the negative 𝑏 over two 𝑎 first. And we’ve already got a common
denominator here, so in fact I’m gonna go one stage further than that. I’m gonna write this all
as one expression on the right-hand side.
And here we have a general formula for solving
quadratic equations of the format 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 is equal to zero. So let’s see
how we can use that in practice. Here’s a question. Solve 𝑥 squared minus five 𝑥 plus six
equals zero. So we’ve gotta find the value or values of 𝑥 which will satisfy that equation.
the first thing to do is to work out what 𝑎, 𝑏, and 𝑐 are from the general formula. So our
general format is 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, so the coefficient of 𝑥 squared is the value of 𝑎.
There’s nothing in front of the 𝑥 squared, so that means there’s one 𝑥 squared, so 𝑎 is equal to
The coefficient in front of the 𝑥 is negative five, so 𝑏 is negative five. And the number
term on its own at the end is positive six, so 𝑐 is positive six. So next we write out our
quadratic formula: 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus
four 𝑎𝑐 all over two 𝑎.
And then we’re gonna substitute in these values to that equation. Now a
couple of top tips: I would always write the equation out in full as we have done here
substituting in all values 𝑎, 𝑏, and 𝑐 exactly as they are. That shows if you’re in exam it
shows the examiner you know basically what you’re doing.
Secondly I would always write negative
numbers in brackets, so we’ve got negative five. I’ve written the negative five in brackets here
and here because 𝑏 was negative five. This means that when you enter the number on your
calculator, your calculator isn’t going to make any mistakes. So for example here, negative five
squared if I’ve just written negative five squared on my calculator that will give me an
answer of negative twenty-five.
But really we want the whole of negative five squared, so
negative five times negative five is positive twenty-five. So this is the answer we’re looking
for. Therefore, the brackets are definitely gonna help us get the right answer. Okay let’s go through
that bit by bit.
The negative of negative five is positive five plus or minus negative five
times negative five is positive twenty-five, so this is gonna be twenty-five, and four times
one is four and four times six is twenty-four, so we’re gonna be taking away twenty-four. And on
the bottom, two times one is just two.
So working out the contents of the square root, there
twenty-five take away twenty-four is one, so we’ve got the square root of one. So the- the answers
are gonna be five plus the square root of one all over two or five take away
the square root of one all over two. So now we write those out separately.
First of all, we’re
gonna do the positive case, then we’re going to do the negative case. So the first instance 𝑥
could be five plus one over two, which gives us an answer of three, or 𝑥 could be five take away one
over two, which gives us an answer of two. Now we just need to write our answer out clearly.
There we go; the answer 𝑥 equals three or 𝑥 equals two. If we wanted to be really sure of
ourselves, we could always try those back in the original equation, so let’s just quickly do
that now. So three, let’s try three first, three squared is nine take away five lots of three, so
we’re taking away fifteen, and then add six, so nine add six is fifteen takeaway fifteen, that’s
zero, that’s correct.
And now with two, two squared is four take away five lots of two, so we’re
taking away ten, and then we’re adding six, well four plus six is ten take away ten is equal to
zero. So we know that’s correct as well, so we- we’re completely happy that our answers are