Video Transcript
The equation of the following graph is 𝑦 equals 𝑥 squared plus five 𝑥 minus 13. Use the iterative formula 𝑥 sub 𝑛 plus one equals 13 over five plus 𝑥 sub 𝑛, starting with 𝑥 naught equals two, to find the positive root of the equation 𝑥 squared plus five 𝑥 minus 13 equals zero to four decimal places.
There’s also a second part to this question which we’ll consider in a moment. When we think about finding roots of an equation, we’re looking for the values of 𝑥, where our graph intersects the 𝑥-axis. The positive location is around here. It looks to be roughly equal to 𝑥 equals two. And that’s why our starting value 𝑥 naught is equal to two. We then have this iterative formula here. It might sometimes feel like these have magically appeared, but all they really are is a rearrangement of the equation we’re looking to solve.
And so what we’re going to do is we’re going to take the first value of 𝑥, that’s 𝑥 naught equals two, and we’re going to substitute it into the formula. When we do, we get the next value of 𝑥 out, so 𝑥 sub one will be equal to 13 over five plus 𝑥 naught. But of course, 𝑥 naught is two. So we find 𝑥 one is equal to 13 over five plus two, which is equal to 1.85714 and so on. We next ask ourselves, “Well, is this value of 𝑥 equal to the previous value of 𝑥 correct to four decimal places?” Well, no, 𝑥 sub naught was equal to two. And so we’re going to continue this process. But this time we’re going to substitute this new value of 𝑥 into our formula, so 𝑥 sub two will be equal to 13 over five plus 𝑥 sub one.
And at this stage, it’s really sensible to use the previous answer button on your calculator. You’ll type in 13 divided by five plus answer. That will be the same as working out 13 divided by five plus 1.857 and so on. That gives us 1.89583 and so on. And we see that whilst it’s equal to 𝑥 sub one correct to one decimal place, it’s not equal to 𝑥 sub one correct to four. So we continue. 𝑥 sub three will be 13 divided by five plus 𝑥 sub two. But of course, we’re using the previous answer button on our calculator, so that allows us just to press equals again without retyping the whole calculation. That gives us 1.88519. Correct to four decimal places, this isn’t equal to the previous value of 𝑥, and so we continue.
We press equals again to get the next value of 𝑥 as 1.88810 and so on. That’s not equal to the previous value of 𝑥. Pressing again gives us 𝑥 sub five is 1.88731. This is still not equal to the previous value of 𝑥. So let’s keep going until we find two subsequent values for 𝑥 that are equal to one another correct of four decimal places. We have to go all the way to 𝑥 sub ℎ to achieve this. Rounding 𝑥 sub seven and 𝑥 sub eight to four decimal places gives us 1.8875 in both cases. And now that we’ve identified two subsequent values of 𝑥 that are equal, we found the positive root to the equation. It’s 𝑥 equals 1.8875. We’re now going to clear some place and consider the second part of this question.
The second part says, use the iterative formula 𝑥 sub 𝑛 plus one equals 13 over 𝑥 sub 𝑛 minus five, starting with 𝑥 naught equals negative seven, to find the negative root of the equation 𝑥 squared plus five 𝑥 minus 13 equals zero to four decimal places.
This is very similar to the first part of this question, but our iterative formula is different. It’s a different rearrangement of the same equation. We’re also going to start with 𝑥 naught equals negative seven. Once again, we’re going to substitute 𝑥 naught into the equation to get 𝑥 one out. We’ll repeat this until subsequent values of 𝑥 are equal to one another to four decimal places. 𝑥 sub one is 13 divided by negative seven minus five. That’s negative 6.85714 and so on. Since this is not equal to the previous value of 𝑥, we continue.
Remember, at this point, we can save some time by replacing 𝑥 sub 𝑛 with the previous answer button. And 𝑥 sub two will be 13 divided by the previous answer minus five, giving us negative 6.89583 and so on. This is not equal to the previous value of 𝑥, and so we’re ready to continue. But of course, we now have the correct calculation set up in our calculator. So we can press equals to get 𝑥 sub three. It’s negative 6.88519, which is not equal to the previous value of 𝑥 correct of four decimal places. Let’s continue until we achieve that.
This time we achieve that when we get to 𝑥 sub seven. When we round each of these values 𝑥 sub six and 𝑥 sub seven correct to four decimal places, we get negative 6.8875. And so we stop here, and we found the negative root of our equation. It’s 𝑥 equals negative 6.8875.
