Video Transcript
The equation of the following graph
is 𝑦 equals 𝑥 squared plus three 𝑥 minus nine. Use the iterative formula 𝑥 sub 𝑛
plus one equals nine over 𝑥 sub 𝑛 minus three, starting with 𝑥 naught equals
negative five, to find the negative root of the equation 𝑥 squared plus three 𝑥
minus nine equals zero to four decimal places.
We’ve been given the graph of a
quadratic equation 𝑦 equals 𝑥 squared plus three 𝑥 minus nine. We’re looking to find the negative
root to the equation 𝑥 squared plus three 𝑥 minus nine equals zero. Well, the root is the value of 𝑥
where the graph intersects the 𝑥-axis. And that simply gives us a solution
to our quadratic equation. Now, looking at the graph, we can
estimate the negative root to be around negative 4.9. But we want a more accurate
result. And so, rather than, say, applying
the quadratic formula or using completing the square, we use something called
iteration.
We’ve been given an iterative
formula and, while these look a little bit scary, all they mean is that we
substitute a value of 𝑥 in and the next value of 𝑥 comes out. So, if we substitute 𝑥 naught into
our equation, 𝑥 one comes out. If we substitute 𝑥 one into the
equation, 𝑥 two comes out and so on. Now, the more we apply this
process, the more accurate solution we get. Now, let’s recall the steps. The first step is to start with an
initial value of 𝑥 naught. Now, one way we could find an
initial value of 𝑥 naught is to find a solution that’s quite close. So, we said 𝑥 is approximately
equal to negative 4.9. However, we’re actually told to use
𝑥 naught equals negative five, which is of course also close. So we’re going to go with that.
Our next step is to substitute the
value of 𝑥 𝑛 into our iterative formula to find 𝑥 𝑛 plus one. Well, our iterative formula is 𝑥
𝑛 plus one equals nine over 𝑥 𝑛 minus three. So, 𝑥 one is nine over 𝑥 naught
minus three. But of course, 𝑥 naught is
negative five. So we get nine divided by negative
five minus three, which is equal to negative 4.8. Our third step is to ask ourselves,
is the value of 𝑥 𝑛 equal to the value of 𝑥 𝑛 plus one to the given degree of
accuracy? In this case, we’re asking, is 𝑥
naught equal to 𝑥 one correct to four decimal places? Well, no. Correct to four decimal places, 𝑥
naught would be negative 5.0000 and 𝑥 one would be negative 4.8000. These are clearly not equal.
If we answer no to this question,
we go back to step two. If, however, we’re able to answer
yes, we stop, and we have our solution for 𝑥. So, let’s go back to step two and
substitute our value for 𝑥 𝑛 into the formula. 𝑥 two will be nine over the value
for 𝑥 one minus three. Of course, we saw 𝑥 one was
negative 4.8. So the calculation we’re going to
do is nine divided by negative 4.8 minus three. At this stage, if you have it, it’s
really useful to use the previous-answer button on your calculator. We’re going to do nine divided by
the previous answer minus three. And this is because we always
substitute the previous answer into this space in our calculation, so it’ll save us
a little bit of time.
𝑥 two gives us a value of negative
4.875 or negative 4.8750. 𝑥 one is not equal to 𝑥 two, and
so we go back to step two and substitute 𝑥 two into our formula. If we’re using the previous-answer
button, we can simply press equals. Otherwise, we need to do nine
divided by negative 4.875 minus three. 𝑥 sub three is negative 4.84615
and so on. Correct to four decimal places,
that’s negative 4.8462. We see this is not equal to our
previous value for 𝑥 two, to negative 4.875. And so we continue. Using the previous-answer button,
we should just be able to press equals, and we get negative 4.85714, which, correct
to four decimal places, is negative 4.8571. 𝑥 three and 𝑥 four are not equal,
so we continue.
We have to go all the way to 𝑥 sub
nine and 𝑥 sub 10 to find two values who round to the same number to the given
degree of accuracy. 𝑥 sub nine and 𝑥 sub 10 are both
negative 4.8541 correct to four decimal places. And so, we go back to our flow
chart and we answer yes to step three, and this means we stop. And so, correct to four decimal
places, the negative root to the equation 𝑥 squared plus three 𝑥 minus nine equals
zero is 𝑥 equals negative 4.8541.
