Video Transcript
Find the values of 𝑥 that
satisfy 𝑥 squared minus three 𝑥 minus ten is less than or equal to zero. So what we’re gonna do is,
we’ll say let’s consider the equation 𝑦 equals 𝑥 squared minus three 𝑥 minus
ten. So we’re basically putting all
of this lot equal to our 𝑦-coordinate. Now this is a quadratic, and
the coefficient of 𝑥 squared is one, so that’s positive. So we know that this would be a
positive happy curve. We also know that the constant
term on the end is negative ten, so 𝐶 is equal to negative ten. And that’s where it cuts the
𝑦-axis. And it cuts the 𝑥-axis when
the 𝑦-coordinate is equal to zero. So since 𝑦 is equal to 𝑥
squared minus three 𝑥 minus ten, what we’re saying is it cuts the 𝑥-axis when
𝑥 squared minus three 𝑥 minus ten equals zero.
And that factors, so 𝑥 squared
minus three 𝑥 minus ten factors to 𝑥 plus two times 𝑥 minus five. And now we’ve got it in the
format. We’ve got something times
something is equal to zero, so one of those things must be equal to zero in
order to get a-a result of that product of zero. So either 𝑥 plus two equals
zero or 𝑥 minus five equals zero. And that means that 𝑥 must be
the equal to negative two to make that equal to zero or 𝑥 must be equal to five
to make that equal to zero.
So now we’ve got enough
information for us to be able to sketch the curve of 𝑦 equals 𝑥 squared minus
three 𝑥 minus ten. Well it cuts the 𝑦-axis at
negative ten, and it cuts the 𝑥-axis at negative two and positive five. So that’s maybe about here,
negative two; and positive five is over here. Now it’s a quadratic, so
that’ll be a symmetric parabola. So the axis of symmetry,
because it’s gonna be midway between negative two and positive five, is gonna be
sort of here somewhere. And the curve is gonna look
something like that. Now as we’ve said at the
beginning, 𝑦 is equal to all of this stuff, and what we’re trying to find is
the 𝑥-values for which that is less than or equal to zero. So we’re looking on this
particular graph for where 𝑦 is less than or equal to zero.
Well 𝑦 is equal to zero here
and 𝑦 is equal to zero here, so negative two and negative five are the
𝑥-values that generate a 𝑦-coordinate of zero. And we’re also looking for the
region for which 𝑦 is less than zero, so that’ll be everything in between. So that’s all the way round
here. So in terms of the 𝑥-values
that generate those 𝑦 coordinates, well 𝑥 is negative two, 𝑥 is five, and
everything in between. They are the valid
𝑥-coordinates. And for the 𝑥-coordinates
we’re not interested in, well look up here, you can see that the 𝑦-coordinate
is greater than zero so we’re not interested in that. So in terms of the region we’re
not interested in, it’s this region out to infinity here; and it’s not including
negative two, but it’s this region out to a negative infinity over here.
So the 𝑥-values we’re looking
for is to generate that 𝑦-coordinate of less than or equal to zero are negative
two is less than or equal to 𝑥 is less than or equal to five. So that’s in inequality
format. In interval format, the ends of
the interval we’re looking for are negative two and five, and they are both
included. So we need to put the square
brackets around those. So that’s in interval
format. And using set notation, we can
say that we’ve got the set of 𝑥 such that 𝑥 is real where negative two is less
than or equal to 𝑥 is less than or equal to five.
So the process that we went
through there was we, first of all, we came up with an equation for 𝑦 equals
some combination of 𝑥, some function of 𝑥, and then we worked out where that
generated a value of zero. And then we were trying to
think of, you know, okay we were looking for the function to be less than or
equal to zero in this case, or it might be equal to zero or greater than zero in
other cases. So you’re then doing those
comparisons. Now the bit that’s really
important that I was talking about at beginning in terms of your working out is
to do this sketch. If you do the sketch, it’s
really clear to see whether you’re looking for points above the 𝑥-axis or
points below the 𝑥-axis. If you don’t do that, lots of
people go through these questions and they-they find out these critical
𝑥-values, but then they just kinda guess at whether we’re going between the
𝑥-values or outside of the 𝑥-values. So this final sketch here is
just really helpful in getting it nice and clear in your mind whether you’re
looking for points for 𝑦-coordinates above the line or below that line, the
𝑥-axis.
