### Video Transcript

Find all the real solutions to the
system of equations ๐ฆ equals negative two ๐ฅ squared plus four ๐ฅ plus six and ๐ฆ
equals four ๐ฅ squared plus 22๐ฅ plus 24.

So when we take a look at our
system of equations, we can see that they both are ๐ฆ equals and then we have an
expression. So therefore, what we can do to
solve the problem is equate them to each other. So therefore, what we can say is
that negative two ๐ฅ squared plus four ๐ฅ plus six is equal to four ๐ฅ squared plus
22๐ฅ plus 24.

So now what we want to do is we
want to actually have a quadratic in the form ๐๐ฅ squared plus ๐๐ฅ plus ๐ equals
zero to help us solve it. So in order to do that, what weโre
gonna do is add two ๐ฅ squared, subtract four ๐ฅ, and subtract six from both sides
of the equation. And when we do that, what weโre
gonna be left with is zero is equal to six ๐ฅ squared plus 18๐ฅ plus 18. And as we can see that each term on
the right-hand side is divisible by six, what we can do is divide through by six to
make it easier to solve. So when we do that, what weโre
gonna get is zero is equal to ๐ฅ squared plus three ๐ฅ plus three.

Or straightaway, we can identify
that this is in fact not going to factor, because we could factor to solve the
quadratic but in this case it wonโt work because we have three. And we need to have a product that
makes three. Well the only product that makes
three is one multiplied by three, which equals three. Well, if you add one and three, you
get four, whereas we want to have a sum of three. So we know that itโs not going to
factor.

Well, weโre now at the stage where
we think how can we solve our quadratic using other methods? What we could do is plug it into
the quadratic equation and have a look at how we get on. And we will do that to show how we
get the same result.

However, what you might notice by
inspection is that the coefficient of ๐ฅ and our numerical term are both positive
and the same value and less than five. And also, itโs worth noting, which
is important, the ๐ฅ squared coefficient is also positive. And by noticing this, we can
actually think, โWell, is it possible that there couldnโt be any real roots at
all?โ

And what we can do to check this
out is use the discriminant, because what we know from the discriminant is that if
we have ๐ squared minus four ๐๐ is less than zero, then there are no real
roots. If ๐ squared minus four ๐๐ is
equal to zero, then thereโs one real root. And if we have ๐ squared minus
four ๐๐ is greater than zero, then there are two distinct real roots.

And in our equation, ๐ is equal to
one because thatโs the coefficient of ๐ฅ squared. ๐ is equal to three because itโs
our coefficient of ๐ฅ. And then our numerical term ๐ is
equal to three as well.

So now what we can do is work out
our discriminant. And to do that, weโll have three
squared minus four multiplied by one multiplied by three. So our discriminantโs gonna be nine
minus 12. So our discriminant is gonna be
negative three. So the discriminant therefore is
less than zero. So we can say that theyโre gonna be
no real roots.

And we managed to notice this
because, as I said, we had the coefficient of ๐ฅ was positive three and the
numerical value was positive three. And we also had the coefficient of
๐ฅ squared being positive. So therefore, what we know is that
it was going to always look like they were gonna be no real roots. Because if we think about it, if we
have a number squared and then we subtract four multiplied by that number and
another number, itโs always gonna be a negative value.

Now this was using the
discriminant. What I did mention was that we
could have found this out also if weโd just gone straight into using the quadratic
formula. Well, the quadratic formula is that
๐ฅ is equal to negative ๐ plus or minus the square root of ๐ squared minus four
๐๐ over two ๐. Well, if we use the quadratic
formula, what weโd have is ๐ฅ equals negative three plus or minus the square root of
three squared minus four multiplied by one multiplied by three all over two
multiplied by one.

So then this would give us ๐ฅ is
equal to negative three plus or minus the square root of negative three over
two. So now what we get is root negative
three in our numerator. Well, root negative three is
undefined, because we canโt have a result for the square root of a negative number
if weโre using real numbers. So therefore, what we could say is
that the system of equations would have no real roots.