Iteration - GCSE Higher Maths

Summary

This educational video introduces the concept of iteration, a method for solving cubic equations that cannot be addressed with the quadratic formula. It demonstrates how to show a solution exists between two values by observing a change of sign in the equation's output. The video then guides through the process of rearranging the equation for iteration, using an iterative formula to approximate the solution progressively closer. It concludes with a practical example, calculating successive terms to find a solution accurate to five decimal places, and verifying the solution's accuracy by substituting it back into the original equation.

Takeaways

• 🔍 The video introduces the concept of iteration, a method for solving equations, particularly cubic equations that cannot be solved using the quadratic formula.
• 📌 Iteration begins by identifying a range where a solution exists, such as between x=1 and x=2, by observing a change of sign when substituting these values into the equation.
• 📈 A change of sign in the equation's output when substituting values indicates a solution exists between those values, which can be visualized on a graph where the curve crosses the x-axis.
• 📝 To apply iteration, the equation must be rearranged so that x is isolated on one side, setting the stage for the iterative formula.
• ✂️ The iterative formula involves taking the cube root of a term involving x, which is derived from rearranging the original equation.
• 🔑 The subscript n+1 denotes the next term in the sequence, while subscript n refers to the current term, allowing for the calculation of successive approximations of the solution.
• 🔄 The iterative process involves substituting the current term into the formula to find the next term, which is then used to find the subsequent term, and so on.
• 📊 The initial term, x0, is given or chosen, and successive terms (x1, x2, x3, etc.) are calculated using the iterative formula.
• 🔢 Iteration can be performed using a calculator, making it easy to compute multiple iterations quickly and accurately.
• 🎯 The goal of iteration is to approximate the solution to a high degree of accuracy, with each iteration bringing the approximation closer to the exact solution.
• 📋 To verify the accuracy of the solution, the calculated value can be substituted back into the original equation, with a result close to zero indicating a precise approximation.

Q & A

What is the main topic of the video?

The main topic of the video is the concept of iteration, specifically how to use it to solve a cubic equation.

Why can't the quadratic formula be used for the given equation?

The quadratic formula cannot be used because the given equation is a cubic equation, with the highest power of x being 3, not 2 as in quadratic equations.

What is the first step to show that a solution to the equation lies between x equals one and x equals two?

The first step is to substitute x with 1 and 2 in the equation and observe the change in sign from negative to positive, which indicates a solution between these two values.

What does a change of sign in the values of the equation when substituting x with 1 and 2 indicate?

A change of sign indicates that there is at least one solution to the equation between the two substituted values, in this case, between x = 1 and x = 2.

How can the iterative formula be derived from the given cubic equation?

The iterative formula is derived by rearranging the equation to make x the subject, then taking the cube root of both sides, and finally writing x with subscripts to represent the iterative process.

What does the subscript 'n+1' represent in the context of the iterative formula?

In the context of the iterative formula, the subscript 'n+1' represents the next term in the sequence of approximations to the solution.

What is the purpose of the iterative formula in solving the cubic equation?

The purpose of the iterative formula is to provide a method to calculate successive approximations to the solution of the equation, with each iteration getting closer to the actual solution.

How is the initial term 'x0' used in the iterative process?

The initial term 'x0' is the starting point for the iterative process. It is used to calculate the first term 'x1', and subsequent terms are calculated using the iterative formula.

What is the significance of calculating x1, x2, and x3 in the iterative process?

Calculating x1, x2, and x3 is part of the iterative process that gradually refines the approximation of the solution to the cubic equation. Each term is closer to the actual solution than the previous one.

How can you determine the accuracy of the solution obtained through iteration?

The accuracy of the solution can be determined by substituting the iterative solution back into the original equation and observing how close the result is to zero. A very small non-zero result indicates a high level of accuracy.

What is the final step in solving the cubic equation using iteration as described in the video?

The final step is to continue the iterative process until the solution stabilizes, and then round the result to the desired number of decimal places, as specified in the question.

Outlines

00:00

📚 Introduction to Iteration for Solving Cubic Equations

This paragraph introduces the concept of iteration as a method for solving cubic equations, which cannot be addressed with the quadratic formula. It demonstrates how to show that a solution exists between x=1 and x=2 by substituting these values into the equation and observing a change of sign from negative to positive. The explanation includes the graphical representation of the equation, where the graph crosses the x-axis, indicating a solution. The paragraph also explains the process of rearranging the equation to isolate x and the introduction of an iterative formula to approximate the solution through successive iterations.

05:00

🔍 Iterative Formula Application and Calculation Process

The second paragraph delves into the application of the iterative formula to calculate the successive terms of the sequence, starting with an initial term x0=2. It explains the iterative process of using the formula to find x1, x2, and x3 by substituting the current term into the formula to find the next. The importance of using the entire number from the calculator to maintain accuracy is emphasized. The paragraph also outlines how to continue the iterative process to approximate the solution to the equation more accurately, eventually reaching a value that, when rounded to five decimal places, gives the solution 1.89329.

Keywords

💡Iteration

Iteration refers to the process of repeating a calculation or algorithm with the aim of approaching a desired result. In the context of the video, iteration is used as a method for solving equations, specifically a cubic equation, by approximating solutions through a sequence of calculations. The script mentions that iteration is particularly useful when the equation cannot be solved using standard algebraic methods, such as the quadratic formula.

💡Cubic Equation

A cubic equation is a polynomial equation of degree three, which means the highest power of the variable (in this case, x) is three. The video script introduces a cubic equation and explains that it cannot be solved using the quadratic formula due to its higher degree. The equation given in the script is an example where iteration is used to find a solution.

💡Change of Sign

The change of sign is a concept used in numerical methods to determine the presence of a solution between two values. In the script, it is mentioned as a way to show that a solution to the cubic equation exists between x=1 and x=2. The script describes how substituting these values into the equation results in a change from negative to positive, indicating a solution in that interval.

💡Graphical Representation

Graphical representation is a visual method of depicting data or functions, such as plotting points on a graph. The video script suggests using a graph to visualize the change of sign, where the points for x=1 and x=2 are plotted, and the graph is expected to cross the x-axis between these points, indicating a solution to the equation.

💡Rearranging Equation

Rearranging an equation involves manipulating the terms to isolate the variable or to set the equation to zero. In the script, the cubic equation is rearranged to make x the subject, which is a necessary step before applying the iterative formula. This process involves adding terms to both sides of the equation to simplify and prepare it for the iteration process.

💡Cube Root

The cube root of a number is a value that, when multiplied by itself three times, gives the original number. In the script, taking the cube root is part of the process to simplify the equation and create an iterative formula. It is used to deal with the cubic term on the left side of the rearranged equation.

💡Iterative Formula

An iterative formula is a mathematical expression that defines the next term in a sequence based on the current term. In the video, the iterative formula is derived from the rearranged equation and is used to calculate successive approximations of the solution. The script explains how to use this formula iteratively to find increasingly accurate solutions.

💡Initial Term

The initial term, or x0 in the script, is the starting point for the iterative process. It is the first value used in the iterative formula to calculate subsequent terms. The video script specifies that x0 equals 2, which is the basis for calculating the values of x1, x2, and x3.

💡Approximation

An approximation is a value that is close to the exact value but not exact itself. In the context of the video, each iteration provides an approximation of the solution to the cubic equation. The script explains that as more iterations are performed, the approximations become closer to the actual solution.

💡Rounding

Rounding is the process of adjusting a number to a specified number of decimal places. In the script, the final step of the iterative process involves rounding the result to five decimal places to provide the solution to the equation with the required precision.

💡Accuracy

Accuracy in mathematics refers to how close an approximation is to the true value. The video script discusses the accuracy of the solution found through iteration by substituting the calculated value back into the original equation and observing the result is very close to zero, indicating a high degree of accuracy.

Highlights

Introduction to the topic of iteration for solving cubic equations.

Explanation of why the quadratic formula cannot be used for cubic equations.

Demonstration of how to show a solution lies between x=1 and x=2 using iteration.

The concept of 'change of sign' indicating the presence of a solution between two values.

Graphical representation of the change of sign on a graph to visualize the solution.

Rearranging the equation to make x the subject for the iteration process.

Use of the cube root to simplify the equation for iteration.

Introduction of the iterative formula and its role in finding successive approximations.

How to derive the iterative formula from the original equation.

Calculating the first three terms of the sequence using the iterative formula.

The iterative process explained as a sequence of calculations to approximate the solution.

Using an initial term x0 and calculating subsequent terms x1, x2, and x3.

The importance of using the entire number from a calculator for accurate iteration.

Solving the cubic equation to five decimal places using iteration.

Technique for using a calculator to perform multiple iterations quickly.

The final solution rounded to five decimal places and its significance.

Verification of the solution's accuracy by substituting it back into the original equation.

Conclusion on the accuracy of the solution and its closeness to zero.

Encouragement for viewers to watch more educational content and subscribe for updates.

Transcripts

in this video we're going to learn about

a topic called iteration

imagine we were trying to solve this

equation here

this equation is known as a cubic

equation because it has an x to the

power 3 as the highest power so it's not

a quadratic equation so we can't use the

quadratic formula for instance

fortunately iteration is a method for

solving equations that will help us

a typical start to an iteration question

might look something like this

part a show that a solution to the

equation lies between x equals one and x

equals two

so the questions actually told us that

one of the solutions is between one and

two

we're asked to show that this is the

case though

to do this all you do is take the

equation and substitute in values x

equals one and x equals two so if we

substitute in one we'd get one cubed

minus two times one minus three which

comes out as negative four

if we substitute in two we get two cubed

minus two times two minus three which

gives us one

at this point we can observe what we

call a change of sign it's a change of

sign because we've gone from negative to

positive it would also be a change of

sign if it went from positive to

negative the reason this shows a

solution is easier to see on a graph

imagine we were sketching this graph and

we were plotting the points for one and

two

when we plotted one it was at negative

four

when we plotted two it was at positive

one so if we were to draw the graph of

this we'd somehow connect up these two

points now it probably wouldn't be a

straight line but somewhere between

these two points it must cross over the

x-axis

when it crosses the x-axis we know that

would be a solution to the equation so a

solution must be between x equals one

and x equals two

so what we can say here is that the

change of sign shows there must be a

solution between the values x equals one

and x equals two

you should note that this method only

works when the equation is equal to zero

if the equation isn't already equal to

zero just rearrange it so that you get

zero on the right hand side then if you

notice a change of sign you know there's

a solution between those two x values

now that we know a solution lies between

1 and 2 we can use iteration to find the

solution really accurately

the first thing we're going to do is

imagine we were asked to make this x the

subject so we're going to rearrange it

so that x is on the left-hand side

to begin we're going to add 2x and 3 to

both sides

if we add this to the left hand side it

cancels out the negative 2x and negative

3 so we're just left with x cubed on the

right hand side 0 add 2x add 3 is 2x at

3.

the next thing to try and do is remove

the cubed

the inverse of cubed is cube root so

we'll cube root both sides

if we cube root the left we just get x

and if we cube root the right we get the

cube root of 2x plus

this might seem like quite an unusual

approach but it will help us out

the next thing to do is take the x on

the left hand side and write it with a

subscript n plus one

and any x's on the right hand side

become x with a subscript n this is now

known as an iterative formula

often the iterative formula is given to

you in the question but you may be asked

to show that it does work like i have

done in this question

the follow-up question could look

something like this

the iterative formula can be used to

solve the equation

part b given that x0 equals 2 calculate

the values of x1 x2 and x3

now to understand how to do this you

need to understand what an iterative

formula is

x m plus 1 here refers to the next term

of a sequence of terms whereas x n

refers to the current term we're on

if for example we started with our

current term as x 0 and we substituted

in that number

and then worked it out it would tell you

what x 1 is

you could then take that term and

substitute it in

work it out and you'd get x2

you could then take that term substitute

it in work it out and you'd get x3 and

you can carry on this iterative process

x3 gives you x4 and so on so all an

iterative formula tells you is the

formula you need to use to work out the

next term and that's what we're going to

do now

in the question we were told that x0

equals 2 this is known as our initial

term

we need to calculate the next three

terms

so we need to calculate x1 first to

calculate x1 we're going to write out

the formula

cube root of 2 times

x0

plus 3 and we're using x zero because

that's the term we currently have

we already know what x zero is though x

zero is two so if we substitute x zero

for two we can calculate this

you can now type this into your

calculator and you'll get this number

here

in an iteration question you're probably

going to need to write your whole

calculator display down

we can now use this value for x1 to get

the next value for x2 so imagine x1 is

our current term

x2 is our next term so we're going to do

the cube root of 2 times our current

term which is x1 and then add 3.

so we substitute x1 for this long number

here

and then calculate it

it's important at this point that you

don't round the long number on your

calculator you want to type in the whole

number or even better use the answer

button on your calculator

so x2 would come out as this number

we can then imagine that x2 is our

current term and then calculate x3

so x3 would be the cube root of 2 times

our current term which is x2 add 3.

we can then substitute the x2 for our

most recent number

which is this one and then type this

into a calculator again being sure to

use the whole number

so x3 would come out as this

we've now completed the question because

we found the values x1 x2 and x3

part c asks us to solve the equation

giving our answer to five decimal places

if we think back this was our original

plan with the question we wanted to

solve the equation

so far we were given the x0 value of 2

and we calculated x1 x2 and x3

each of these successive iterations is

an approximation to the solution the

further you go the better the answer so

to find the solution we need to keep

this going we can find x4 x5 and so on

fortunately this is made a lot easier

using a calculator

we can calculate iterations really

quickly using a calculator to do this

i'm going to press 2

and then equals because 2 was the

initial value of x0 that was given to us

in the question the effect of this is

this number 2 here is stored in the

calculator's memory it was the most

recent number it calculated

now we'll go ahead and do the iterations

so we need to do the cube root because

it started with a cube root that's on

shift so shift cube root that it was two

times the previous answer so i know the

previous answer x0 that was 2 so i could

just press another 2 at this point or i

could press the answer button because

that's the number the calculator most

recently calculated

then it was plus 3 and then press equals

and you can see the number on screen

matches the one that we calculated

now we need to get x2 and you could do

it by repeating all of those steps but

the calculation you need is already on

the screen if we simply press the equals

button now the calculator will take the

most recent answer the one displayed on

screen and put it into this formula here

so if we press equals you'll see we get

the next answer which is x2 and then we

press equals

x3

equals

x4

x5 x6 x7 x8 x9 and so on

in fact you can go even further than

this if you continue to press equals

you'll see the number gets closer and

closer to the exact answer and

eventually when you get to this point

here no matter what you do if you keep

pressing equals the number has settled

so this is the number we need to round

to get our final answer the question

said five decimal places so one two

three four five

so the fifth decimal point here is an

eight but it's got a 9 after it so we'll

need to round up so the final answer

rounded to 5 decimal places is 1.89329

so we've managed to find one of the

solutions to this equation

a common follow-up question is to ask

you just how accurate your solution is

it might read something like this

part d by substituting your answer from

part c into x cubed minus 2x minus 3

comment on the accuracy of your solution

to the equation x cubed minus 2x minus 3

equals 0.

all we need to do here is write out the

equation again but instead of x we're

going to write our solution 1.89329

1.89329 cubed minus two lots of one

point eight nine three two nine minus

three

now if you type this into a calculator

you'll get zero point zero zero zero

zero zero seven zero three five two five

so a very very small number

so we could say that this is very close

to zero so our solution is a good

estimate for the real solution

thank you for watching this video i hope

you found it useful check out what i

think you should watch next and also

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