### Video Transcript

Show that there is a solution to
the equation π₯ cubed plus nine π₯ minus four equals zero between π₯ equals zero and
π₯ equals one.

To answer this question, we need to
evaluate the function π₯ cubed plus nine π₯ minus four at both π₯ equals zero and π₯
equals one. When π₯ is equal to zero, we have
zero cubed plus nine multiplied by zero minus four, which is equal to negative
four. When π₯ is equal to one, we have
one cubed plus nine multiplied by one minus four. This gives one plus nine minus four
which is six.

The key point here is that the
function π₯ cubed plus nine π₯ minus four takes a negative value when π₯ is equal to
zero and a positive value when π₯ is equal to one. As π₯ cubed plus nine π₯ minus four
is a continuous function, which means there are no gaps, this means that it must be
equal to zero somewhere between π₯ equals zero and π₯ equals one as its value has
crossed over from negative to positive between these values. This method of showing us that the
equation has a solution between two values is sometimes called the change of sign
method.

Rearrange the equation π₯ cubed
plus nine π₯ minus four equals zero into the form π₯ equals π over π₯ squared plus
π.

Weβve been asked to rearrange the
equation from a form where the highest power of π₯ is a three to a form where the
highest power is a two. This suggests that we should
factorize by π₯. However, the third term negative
four doesnβt have a factor of π₯. So we can only factorize from the
first two terms. This gives π₯ multiplied by π₯
squared plus nine minus four is equal to zero.

Now, this is starting to look a
little bit more like what weβre hoping to rearrange to as we have a bracket π₯
squared plus nine, which looks like the denominator of the fraction. Next, we can add four to both sides
of the equation, giving π₯ multiplied by π₯ squared plus nine is equal to four.

The final step is to divide both
sides of the equation by that bracket β π₯ squared plus nine β giving π₯ is equal to
four over π₯ squared plus nine. Now, we arenβt explicitly asked to
state the values of π and π. But comparing our equation to the
form we were asked for, we can see that the value of π is four and the value of π
is nine.

Use the iterative formula π₯ π
plus one is equal to π over π₯ π squared plus π with the values of π and π
found above to estimate a solution to π₯ cubed plus nine π₯ minus four is equal to
zero. Start with π₯ zero equals zero and
use the iterative formula three times. Give your answer to three decimal
places.

First, letβs recall our answer to
the previous part of the question, which was that π₯ is equal to four over π₯
squared plus nine. This means that the value of π in
our iterative formula is four and the value of π is nine.

Now, letβs recall what an iterative
formula means. Itβs used to estimate the solution
to an equation by progressively improving the estimate. It tells us that to find the next
value of π₯, π₯ π plus one, we just substitute the current value of π₯, π₯ π, into
this formula. As this equation is a rearrangement
of the equation π₯ cubed plus nine π₯ minus four equals zero, the values of π₯
generated by this iterative formula will give an estimate of a solution to the
equation π₯ cubed plus nine π₯ minus four equals zero.

Weβre told that the starting value
for our estimate π₯ zero is equal to zero. To find the value of π₯ one, we
substitute the value of π₯ zero into the iterative formula, giving four over zero
squared plus nine. This is equal to four-ninths or as
a decimal itβs equal to 0.4 recurring.

To find the next value of π₯, π₯
two, we substitute our current value of π₯ into the formula, giving four over
four-ninths squared plus nine. As a decimal, this gives 0.434899
and the three dots indicate that the decimal continues. To perform this accurately on your
calculator, particularly, if youβre using the iterative formula several times, you
can use the previous answer button.

If you use the fraction and
brackets keys on your calculator, you can input four over answer squared plus
nine. And your calculator will then take
the value on screen, square it, add nine, and then divide four by this. This means that you can just keep
pressing enter and your calculator will keep giving you the next value in this
iterative sequence.

Weβre told to use the iterative
formula three times. So we need to perform one more
iteration. The calculation for π₯ three is
four over 0.4348 squared plus nine. But in reality, you can just press
equals on your calculator if youβve used the previous answer button. This gives a decimal of 0.43529656
and the decimal continues.

Weβre asked to give our answer to
three decimal places. And as the deciding number β so
thatβs the fourth decimal place β is a two, we round down. Our estimate of a solution to the
equation π₯ cubed plus nine π₯ minus four equals zero to three decimal places is
0.435.

Substitute your value for π₯ three
into π₯ cubed plus nine π₯ minus four. Comment on how accurate your
estimate is to the solution of π₯ cubed plus nine π₯ minus four is equal to
zero.

Our estimate of π₯ three is
0.435. So substituting this into the given
expression, we have 0.435 cubed plus nine multiplied by 0.435 minus four. Evaluating this on a calculator
gives an answer in standard form: negative 2.687125 multiplied by 10 to the power of
negative three.

Multiplying a number by 10 to a
negative power means that the number is getting smaller. In this case, it means that we are
dividing by 10 to the power of three or 1000. Dividing negative 2.687125 by 1000
gives the number negative 0.002687125, which has its decimal point three places left
of where it was in the original number.

Weβre asked to comment on how close
our estimate is to the solution of π₯ cubed plus nine π₯ minus four is equal to
zero. Since the value we get when we
substitute our estimate into this expression is very close to zero, this means that
our value of π₯ three is close to the solution.