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Hello, my name is Mr. Clasper and today we are going to be solving equations by using iteration.

We can solve the equation x squared minus two x equals one by using an iterative formula.

Reading this formula from left to right, this means to find the next value for x, we need to find the square root of one plus two lots of the current value for x.

If we're told that x zero is equal to two, and we need to find the values of x one and x two, we need to substitute x zero into the iterative formula on the right hand side, to find the value of x one.

So if we calculate the square root of one plus two lots of our x zero, which is two, we get a value of 2.

236068.

And to find the value of x two, we need to take our new value and substitute that into the right hand side of our iterative formula.

This would give us a value of 2.

33926.

If we wanted to keep going, we could find the value of x three.

So we could take our value of x two and substitute this into the formula on the right hand side.

If we want the value of x four, we take x three and substitute this into the iterative formula.

If we want x five, we repeat the process with x four and so on.

If we continue the process, we will find that the values returned will vary very little.

If we look towards the end of our table, we can see that the value stays constant at 2.

414213.

So this means we have an approximate solution of 2.

414213.

Let's see how we can use our calculators efficiently for this process.

The first thing we need to do is to make sure our calculator remembers the value of two.

This will become more obvious when we get on to step two.

So your first step is to take your value for x zero, and then press equals.

So at this moment in time if you follow this step, your calculator will currently remember the value of two.

The next step involves the use of the Answer button.

The Answer button remembers the last answer, which you calculated.

So as of this moment in time, if you followed our steps, your calculator is currently remembering the value of two.

So whenever you use the Answer button, this is equivalent to using the value of two.

So if we press the square root of one plus two, multiplied by answer, your calculator thinks you want to calculate one plus two multiplied by two.

And pressing equals once will return our value of 2.

236067977 which is our value of x one.

If you follow those steps, your calculator currently is remembering the number 2.

236.

So if you press equals one more time, it will substitute this value where you have the answer in your equation on your calculator display.

This will give you the value of 2.

414213562 as your value for x two.

And if you continue to press equals, you will get closer and closer to a final solution.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So remember, your first step should be to put your first value into the calculator which in this case is one.

Get your calculator to remember this then set up your equation So in your calculator, you want the square root of nine plus five, lots of brackets, A and S.

And when you do this, it will help you find the first two values from the first two iterations.

For part B, if you continue to iterate, you should eventually get a number which rounds to 6.

41.

Here's a question for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So again, remember, you need to set up your calculator to remember the number three first, once you've done that, set up your equation using the A and S button, and you should get an approximate answer of 4.

92.

And when you do this with negative three as a starting value, you will also find that you get 4.

92.

So it doesn't matter how far away you are from this, it will eventually converge to an approximate solution.

Here are some questions for you to try.

Pause the video to complete your task and click resume once you're finished.

And here are your solutions.

So for the first part, our first step would be to divide by x, this leaves x squared minus two is equal to one over x.

From here, we can add two to both sides.

And our final step is to take the root of this.

And for part B, you just need to make sure that we use the appropriate notation so that we have an iterative formula.

Here are part C, and D Pause the video to complete your task and click resume once you're finished.

Anterior solutions, so for part C, if we substituted negative nought point two, we would end up attempting to square a negative value, which we can't do.

And for part D, if you have the same equation setup, and you start with 0.

1, you should eventually get a solution which is approximately 1.

62.

Here's your last question.

Pause the video to complete your task and click resume once you're finished.

And here is your solution for question four.

So remember, to start the process, if you make sure your calculator remembers the value of four, then you need to input the square root of five over answer, plus answer.

And if you continue to press equals, you should get a solution of approximately 2.

116.

And that brings us to the end of our lesson.

So now you can solve equations by using iteration.

Why not give the exit quiz ago to show off your new skills.

I'll hopefully see you soon.