Teaching Pack: Cambridge International AS & A Level Mathematics 9709
Teaching Pack: Cambridge International AS & A Level Mathematics 9709
Teaching Pack: Cambridge International AS & A Level Mathematics 9709
1.1 Quadratics
Cambridge International AS & A Level
Mathematics 9709
Version 1
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Contents
Contents ............................................................................................................................................................ 3
Introduction ........................................................................................................................................................ 4
Lesson preparation ............................................................................................................................................ 5
Lesson plan 1: Completing the square ............................................................................................................. 8
Lesson plan 2: Solving quadratic equations ................................................................................................... 11
Lesson plan 3: Sketching quadratic functions ................................................................................................ 14
Lesson plan 4: Solving quadratic inequalities ................................................................................................. 18
Planning your own lessons .............................................................................................................................. 22
Lesson reflection.............................................................................................................................................. 23
Worksheets and answers ................................................................................................................................ 24
Teacher preparation
Lesson plan
Lesson resource
Lesson reflection
Video
Teaching Pack: 1.1 Quadratics
Introduction
This pack will help you to develop your learners’ skills in mathematical thinking and mathematical
communication, which are essential for success at AS & A Level and in further education.
Each Teaching Pack contains one or more lesson plans and associated resources, complete with
a section of preparation and reflection.
Each lesson is designed to be an hour long but you should adjust the timings to suit the
lesson length available to you and the needs of your learners.
Important note
Our Teaching Packs have been written by classroom teachers to help you deliver
topics and skills that can be challenging. Use these materials to supplement your
teaching and engage your learners. You can also use them to help you create
lesson plans for other topics.
This content is designed to give you and your learners the chance to explore a more active way
of engaging with mathematics that encourages independent thinking and a deeper conceptual
understanding. It is not intended as specific practice for the examination papers.
The Teaching Packs are designed to provide you with some example lessons of how you might
deliver content. You should adapt them as appropriate for your learners and your centre. A single
pack will only contain at most five lessons, it will not cover a whole topic. You should use the
lesson plans and advice provided in this pack to help you plan the remaining lessons of the topic
yourself.
Lesson preparation
This Teaching Pack will cover the following syllabus content:
The remaining two bullet points for topic 1.1 Quadratics are not covered in this Teaching Pack (see
the syllabus for detail). You will need to write your own lesson plans for these items.
When planning any lesson, make a habit of always asking yourself the following questions about
your learners’ prior knowledge and skills:
Learners will need to be confident with the terminology used for algebra and algebraic equations
from their IGCSE (or equivalent) course. This terminology is also identified below.
If learners cannot derive and solve simultaneous equations in two unknowns they will be at a
disadvantage when building on these ideas when solving a pair of simultaneous equations of which
one is linear and one is quadratic.
Learners will need to be able to expand products of algebraic expressions and factorise quadratic
expressions, otherwise they will be at a disadvantage when using this knowledge to solve all types
of quadratic equations and quadratic inequalities. For quadratic equations in different variables,
learners will need to be able to solve simple trigonometric equations e.g. tan 𝑥𝑥 = 3.
Even if learners had mastered the areas above during their IGCSE (or equivalent) course, it is best
not to assume that they are still fluent in this topic as it can lead to learners struggling to solve
problems at AS and A Level.
• All quadratic expressions can be put in the form 𝑎𝑎(𝑥𝑥 + 𝑏𝑏)2 + 𝑐𝑐 and we can use this form to
sketch quadratic functions.
• The discriminant of a quadratic polynomial allows you to determine how many real roots a
quadratic equation will have.
• You can use a sketch of a quadratic function to determine the solution set for a quadratic
inequality in one variable.
• You can use the techniques for solving quadratic equations when an equation can be
rearranged into the form 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0 where 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 are real numbers.
quadratics in some function of 𝒙𝒙 any expression of the 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 where 𝑎𝑎, 𝑏𝑏, 𝑐𝑐 are
real numbers and 𝑥𝑥 is a function of 𝑥𝑥.
Insights video
There is an Insights video linked to this Teaching Pack:
• 1.1 Quadratics – watch this video which will show you how to help your learners to better
understand how to solve quadratic equations using the ‘completing the square’ technique.
Teacher tutorials
There are four sets of Teacher tutorials linked to this Teaching Pack:
• Completing the square – review this tutorial before teaching Lesson plan 1; this will show
you how to highlight the important parts of the completing the square process.
• Solving quadratic equations – review this tutorial before teaching Lesson plan 2; this will
show you how to determine whether an equation is a quadratic in a function of 𝑥𝑥.
• Sketching quadratic functions – review this tutorial before teaching Lesson plan 3; this
will show you how to highlight the important features required to sketch a quadratic
function.
• Solving quadratic inequalities – review this tutorial before teaching Lesson plan 4; this
will show you how to connect solving a quadratic inequality with sketching a quadratic
function.
Lesson progression
Lesson 1 covers completing the square. Lesson 2 focuses on the solving of quadratic equations.
The content from these two lessons is then used in Lesson 3. Lesson 4 builds on the ideas
explored in all three lessons and applies them to solving quadratic inequalities.
Going forward
This topic links to all aspects of the syllabus content through the solving of quadratic equations and
inequalities.
Dependencies
Learners need to know how to expand perfect squares e.g. expand (𝑥𝑥 + 3)2 . Learners will need
their algebraic manipulation skills from IGCSE (or equivalent).
Common misconceptions
Problems this can An example way to resolve the
Misconception
cause misconception
Expanding (𝑥𝑥 + 𝑏𝑏)2 incorrectly If learners believe this Using cognitive conflict should
e.g. (𝑥𝑥 + 3)2 = 𝑥𝑥 2 + 9. then they will not be help learners be more aware.
able to engage with the Using the image of a square of
completing the square length 𝑥𝑥 + 𝑏𝑏 and asking how
process correctly. many terms are generated by the
calculation (𝑥𝑥 + 𝑏𝑏)2 may help
learners break their
misconceptions.
Timings Activity
Starter/Introduction
Worksheet A: Completing the square (squares/algebra)
Give out Worksheet A. Learners work on Part 1, or display this with a projector and let
learners use mini whiteboards. This will allow you to review each learner’s work
quickly.
This activity should be a refresher on expanding (𝑥𝑥 + 𝑏𝑏)2 and enable learners to
make connections to the language of a perfect square.
Timings Activity
• Why do you think we use the phrase ‘a perfect square’ to describe this set of
quadratics?
• Looking at your answer, can you tell what the ‘side length’ of the square will
be when the quadratic is in the form 𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐?
Learners should be able to expand each bracket. Some learners may have forgotten
how to or have the misconception highlighted above. You could use the example way
to resolve the misconception to aid both issues in this case.
Now ask learners to work on Worksheet A Part 2. Ask: Are there any connections
between the two parts?
Learners should be able to link the previous part on perfect squares to filling in the
required information to answer Part 2.
Support: Give another example to aid learners. Start with 𝑥𝑥 2 + 6𝑥𝑥 + □ and ask the
same two questions. This part links directly with Part 1 Question 1.
Challenge: Ask learners to try Part 3. What does this question say about all
quadratic expressions?
Main lesson
Lesson slides: Completing the square
Worksheet B: Completing the square
Play the video on slide 2 and use Worksheet B. Ask learners to describe what is
happening.
Pause the video at about 47 seconds which is the end of (i) – then review the
questions below before moving to the next section of the video. Pause again at 1min
47 and repeat the process.
Example questions to ask as you pause the video after each new slide:
• What is being identified first?
• What are the steps in this procedure?
• Which terms appear to be the most important?
Collect learners’ ideas on a board visible to the whole class.
Learners should begin to identify that the most important terms are the 𝑥𝑥 2 and 𝑥𝑥
terms. They should also be able to spot that (for the case of 𝑥𝑥 2 coefficient 1) we are
halving the value of the coefficient of the 𝑥𝑥 term.
Timings Activity
• Are there any questions that are not like the quadratics we have seen so far?
Learners should be able to complete questions 1 and 2 without many issues.
Learners will find questions 3, 5 and 6 more challenging as the coefficient of 𝑥𝑥 is odd.
Learners will find questions 4, 7 and 8 most challenging and may require the support
of other learners.
Challenge: Ask learners to work on questions 4, 7 and 8, and write down their
solution as a guide to help other learners with this process.
After 15 minutes ask learners to collect their solutions together in pairs and share
their work.
Lesson slides: Completing the square
Show learners the slide 3. Ask the pairs to make any corrections they need to and
highlight the errors they made. Collect the main ideas and issues each pair finds and
highlight them to the class.
Plenary
Worksheet E: Completing the square – spot the errors
Use this task to get learners to review their learning by spotting common errors and
misconceptions. Use ‘Think-pair-share’ but the pair will share with another pair, rather
than the whole class (see How to engage your learners guide for details of this
technique). Each time an expression is discussed one learner is chosen to highlight
the error or misconception. Circulate the room and listen to each group. Handout the
answers at the end and make sure each group of four has seen and understood the
errors.
Reflection Reflect on your lesson; use the Lesson reflection notes to help you.
Dependencies
Learners need to remember the content on quadratic equations from IGCSE (or equivalent) and
how to complete the square.
Common misconceptions
An example way to resolve the
Misconception Problems this can cause
misconception
There is a difference Learners may try and solve Give learners a selection of quadratic
between factorising an expression or only expressions and equations. Ask learners
and completing the factorise a quadratic what process you can perform on each
square on an equation and not find the object.
expression, and values of 𝑥𝑥.
solving using each of
these processes.
The a, b and c Not identifying a, b and c Give learners a selection of quadratic
coefficients can be correctly will lead to equations written in completely different
linked to any term in incorrect solutions. ways. Ask learners to rearrange the
the quadratic without questions into the correct form and
affecting the solution identify
using the quadratic a, b and c.
formula
Timings Activity
Starter/Introduction
Worksheet C: Quadratic expressions
Give a piece of A3 paper to each pair of learners. Using Worksheet C, learners in
pairs factorise questions 1 to 4 and complete the square on questions 5 to 8. They
write these expressions down on their A3 piece of paper.
Learners should be able to access this introduction as they will have recently
completed a lesson on completing the square.
Main lesson
Use the expressions learners have from the introductory activity. Ask ‘What can we
do to turn each of these expressions into equations?’ Highlight that we need each
expression to be equal to something e.g. a number, another expression etc.
In this case, we will let each expression equal zero and then solve. Model question 1
and question 8 with learners. Be aware that question 7 has no solutions. Ask learners
to complete their questions in pairs on their A3 sheet of paper.
Choosing an equation solved for each form, learners can then collect this information
and write their own worked example with notes highlighted in the class discussion.
Support: Some learners may need some additional support. If you see this,
encourage pairs to come together to form a stronger group of learners who can
assist each other.
and asking learners to identify a, b and c in each case. Highlight that the quadratic
equation must be written in this form 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0 in order to use the quadratic
formula e.g. ask learners to solve 2𝑥𝑥 2 = 3𝑥𝑥 − 1.
Challenge: Ask learners to complete the square on 𝑎𝑎𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0 and derive
the quadratic formula.
Timings Activity
in 𝑥𝑥 we have as our quadratic equation. Ask them to rearrange the equations so that
they can be solved e.g. replace 𝑥𝑥 2 with 𝑌𝑌.
After 10 minutes, ask learners to try and solve the quadratic equations they have
formed.
Circulate the room and discuss this activity with learners: How did you decide which
function was a quadratic equation?
Plenary
Ask learners to identify a quadratic equation that they would give as an example for:
• factorising as a process to help solve
• completing the square as a process to help solve
• the quadratic formula.
Challenge: Can learners identify from the coefficients a, b and c if a quadratic
equation will factorise? Can learners identify from the coefficients a, b and c if a
quadratic question can be solved?
Reflection Reflect on your lesson; use the Lesson reflection notes to help you.
Dependencies
Learners need to know how to factorise and complete the square on quadratic functions.
Common misconceptions
An example way to resolve the
Misconception Problems this can cause
misconception
Learners only focus on If learners believe this then From algebra to geometry: using a
the x-intercepts (roots) they will not be able to graphing package, such as Desmos,
of the quadratic sketch or identify stretches and varying k for y=k(x+1)(x-4). Ask
function when of quadratic functions. learners to describe what they see.
sketching. From geometry to algebra: using
Worksheet G in this lesson can enable
learners to understand the need for
another point of the parabola in order to
determine stretches.
Timings Activity
Starter/Introduction
Using mini whiteboards or other writing materials, ask learners to sketch 𝑦𝑦 = 𝑥𝑥 2 .
Highlight to learners that sketching is not the same as plotting. When we sketch a
graph we need to highlight the key features: shape, intersects, turning points.
Learners will need to recall this basic shape from IGCSE (or equivalent). They may
need some support in doing this.
Timings Activity
Support: Ask learners to create a table of values for 𝑦𝑦 = 𝑥𝑥 2 if they are not able to
recall the shape of the graph. What are the key features of this graph? Do we need
to plot all the points to show these key features?
Ask learners about the shape they have created. Define this shape as a parabola.
1) What value of 𝑥𝑥 will give 𝑦𝑦 = 0? What does this mean for the position of the
parabola?
Ask learners what form part 4, 𝑦𝑦 = (𝑥𝑥 − 2)2 − 9 takes. Highlight to learners that this is
in a completed square form and it allows them to ‘see’ where the minimum of this
parabola is. We call this point (and the corresponding maximum point on negative
coefficients of 𝑥𝑥 2 ) the vertex of the parabola. Make it explicit that (𝒙𝒙 − 𝟐𝟐)𝟐𝟐 will
always be greater than or equal to zero. Therefore, when 𝑥𝑥 = 2 this will give the
minimum value of zero.
Support: Learners will find this challenging if they did not meet transformations
explicitly in IGCSE (or equivalent). Encourage learners who are confident to help
other learners with their understanding. Allow them to move around the room to
help each other. This will allow you to listen to their language and make any
suggestions about how they are interpreting the sketching of the graphs.
Timings Activity
Main lesson
Lesson slides: Sketching quadratic functions
Display slide 3 Which parabola? Ask learners to read the instructions and then to
work in pairs and determine the equations of the parabolas.
After 5 minutes, bring the class together to share one of the ideas they have used or
discovered so far. Collect these ideas on a board visible to the whole class. You can
use the Which parabola? teachers notes (included in the Lesson slides notes) to help
structure this part of the lesson.
Support: Some learners will initially find the image overwhelming and will require
more confident learners to help them persevere. You could give learners a hint: We
have already looked at a method of ‘seeing’ where the graph of a quadratic function
will be in the 𝑥𝑥𝑥𝑥-plane by using a form for its algebraic equation.
Allow learners then to continue for another 5 minutes. Then ask learners to meet up
with a second pair and evaluate their solution and their strategy for finding the
solution. Share findings as a class.
Support: Ask learners to complete the square on the two equations. Where might
these graphs be, given their vertex?
Learners will most likely now focus on the vertex of the parabola, whose coordinates
are not marked on the graph.
Support: If you know the roots of the parabola, where will the 𝑥𝑥-coordinate of the
vertex be? How can you use this information to find an equation for the parabola?
If you know the roots of the parabola, how else can you write an equation for the
quadratic function?
Some learners may spot that you can find all four equations by setting up a series of
simultaneous equations for each parabola, using the three points and the general
equation for a quadratic function 𝑦𝑦 = 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 to find a, b and c. Highlight to
learners that this will work, but is inefficient.
After 15 minutes, collect ideas together from the class. Split the responses into ideas
that are specific to the values used in the graph and bigger ideas that could be used
for any problem of this form.
Challenge: Learners can attempt graphs C and D. Learners can also sketch the
quadratic functions given on Worksheet C.
Plenary
Timings Activity
Give out the A6 paper to all learners. Ask learners to create their own problem based
on the ideas in this lesson on one side of the A6 paper.
You could give the following list of concepts for learners to choose from: using
completing the square, using factorisation, creating a new parabola given an equation
of a similar parabola, finding an equation of a parabola given a graph.
Ask learners to write down their solution on the back of the paper.
Collect in the A6 pieces of paper and use as class retrieval practice in a future lesson.
Reflection Reflect on your lesson; use the Lesson reflection notes to help you.
Dependencies
Learners need to know how to solve quadratic equations (Lesson plan 2) and sketch quadratic
functions (Lesson plan 3). Learners should also be familiar with solving linear inequalities.
Common misconceptions
An example way to resolve the
Misconception Problems this can cause
misconception
Inequality signs are These misconceptions Using the introductory activity, you can
interchangeable might exist before starting highlight the correct use of the inequality
this lesson/material. This symbols.
Inequality signs are not will make building on Giving a simple statement, such as 6 <
previous learning
affected by 7, and asking learners to multiple/divide
challenging in order to
manipulations involving by negative one. Any incorrect solutions
solve quadratic inequalities.
negative numbers. learners provide will result in some
cognitive conflict at seeing the incorrect
statement, such as -6 < -7.
All inequality solution If learners believe this they Allow learners to say in words first what
sets are written as one will not be able to find the the set of solutions is, e.g. x is less than
combined inequality in correct solutions sets -1 but greater than 4. Ask the question,
x. where there are two is it possible to be the same value of x
e.g. −1 > 𝑥𝑥 > 4 regions. that satisfies both of these conditions?
Timings Activity
Starter/Introduction
Lesson slides: Quadratic inequalities
Worksheet H: Review of linear inequalities
From the images on slide 3-6/Worksheet H, ask learners to write down on mini
whiteboards or other writing materials the correct inequalities in x which describe the
sets of numbers identified on the number line, then choose learners to share their
solutions with the class.
The purpose of this activity is to review inequality notation and symbols.
Learners may notice that the solution sets for Worksheet A are the answers to the
inequalities on Worksheet H.
From Worksheet H, ask learners in pairs to solve the linear inequalities and write
down the solution sets. The purpose of the activity is to draw out any misconceptions
learners may have when it comes to the use of inequality signs.
You could confirm this to one pair and allow this information to filter to other pairs.
This gives the pairs of learners a chance to confirm their own answers to Worksheet
H with other pairs.
Some learners may have made errors based on misconceptions of inequalities from
previous learning.
Knowing what the answers need to be will help learners to look back over their
answers in their pairs and check with another pair to determine the error in their
steps. This will most likely be poor use of inequality signs or issues with negative
numbers. Learners will see the error and correct it themselves, with the help of their
peers.
Main lesson
Lesson slides: Quadratic inequalities
Worksheet I: Quadratic graphs
Display slides 6-9, or give out hard copies of Worksheet I to each pair of learners.
Each image consists of a parabola, a straight line and an identified region on the x-
axis. Take two minutes and make a note in your pairs of anything you notice about
each image.
Learners will seem a little confused at first as there are four images containing a lot of
information.
Support: You could add some values to the x-axis of image A and ask learners to
explicitly state what has been highlighted on the x-axis. You could ask the follow-up
question:
Timings Activity
What output values do we get from our quadratic function if we input these x-
values?
After five minutes, join pairs together to form groups of four. Ask learners to compare
what they identified and come up with a list of things they noticed and write this down.
Learners will give a range of things they have noticed. They may include:
There is a parabola (quadratic graph) intersecting the x-axis.
There is a straight line intersecting a parabola.
There is a set of values on the x-axis that are identified using the open and closed
dots we have seen for inequalities.
The red line has something to do with the intersections of the parabolas with various
lines.
As you circulate the room, ask students to provide ideas that are interesting for the
purpose of showing a solution set for a quadratic inequality involving a parabola and
a straight line.
Use learners’ ideas of what they have noticed to draw their attention to what each
image is showing.
Image A: Here we have a parabola intersecting the x-axis. The x-values which give
y-coordinates less than or equal to zero are identified.
Image B: Here we have a parabola intersecting a straight line. The x-values which
give
y-coordinates of the parabola greater than or equal to y-coordinate of the line are
identified.
Image C: Here we have a parabola intersecting a straight line. The x-values which
give
y-coordinates of the parabola greater than the y-coordinate of the line are identified.
Image D: Here we have a parabola intersecting a straight line. The x-values which
give
y-coordinates of the parabola less than the y-coordinate of the line are identified.
Challenge: Create equations for each of the graphs in images B, C and D. Use
these equations to help describe inequalities whose set of x-values that satisfy
each are the identified red regions on the x-axis.
Timings Activity
Try and suggest another parabola equation for Image B and this time an equation
for the straight line. The set of x-values identified are when the parabola’s y-
coordinate is greater than or equal to the y-coordinate of the line, what would that
look like as an inequality using your examples?
Support: Select questions (a), (d) and (e) to do first, as they are inequalities in x.
Did their original suggestion for the type of solution set for each inequality make
sense when compared to their final answer? If not, where could the error be? Where
is it most likely to be?
Reflection Reflect on your lesson; use the Lesson reflection notes to help you.
Follow the structure of the Teaching Pack, and use techniques from the ‘How to’ guides, to create
your own engaging lessons to cover these bullet points. Consider what preparation you need for
each lesson: what prior knowledge is needed, what are the key objectives, what are the
dependencies, what common misconceptions are there, and so on.
Below, we have provided an outline of some activities and approaches you might like to try.
You may also find it useful to use the Underground Mathematics Quadratics Pick a card activity. to
review the main concepts on the topic of quadratics.
You will find some other activity suggestions in the Scheme of Work.
Lesson reflection
As soon as possible after the lesson you need to think about how well it went.
One of the key questions you should always ask yourself is:
Did all learners get to the point where they can access the next lesson? If not, what will I do?
Reflection is important so that you can plan your next lesson appropriately. If any misconceptions
arose or any underlying concepts were missed, you might want to use this information to inform
any adjustments you should make to the next lesson.
It is also helpful to reflect on your lesson for the next time you teach the same topic. If the timing
was wrong or the activities did not fully occupy the learners this time, you might want to change
some parts of the lesson next time. There is no need to re-plan a successful lesson every year, but
it is always good to learn from experience and to incorporate improvements next time.
To help you reflect on your lesson, answer the most relevant questions below.
Summary evaluation
What two things went really well? (Consider both teaching and learning.)
What two things would have improved the lesson? (Consider both teaching and learning.)
What have I learned from this lesson about the class or individuals that will inform my next lesson?
Worksheet Answers
C: Quadratic expressions 27 38
C: Quadratic expressions 27 39
Part 1:
1) Expand (𝑥𝑥 + 3)2
Part 2:
1) Given
𝑥𝑥 2 + 10𝑥𝑥 + =
What number must be hidden for the expression to be a perfect square? What are the dimensions
of the square?
2) Given
𝑥𝑥 2 + 8𝑥𝑥 + =
What number must be hidden for the expression to be a perfect square? What are the dimensions
of the square?
3) Given
𝑥𝑥 2 − 8𝑥𝑥 + =
What number must be hidden for the expression to be a perfect square? What are the dimensions
of the square?
Part 3:
1) Find the values of A and B for which 𝑥𝑥 2 − 12𝑥𝑥 + 5 ≡ (𝑥𝑥 + 𝐴𝐴)2 + 𝐵𝐵
3) Find the values of A, B and C for which 2𝑥𝑥 2 − 12𝑥𝑥 + 5 ≡ 𝐴𝐴(𝑥𝑥 + 𝐵𝐵)2 + 𝐶𝐶
4) Find the values of A, B and C for which 4𝑥𝑥 2 − 12𝑥𝑥 + 5 ≡ 𝐴𝐴(𝑥𝑥 + 𝐵𝐵)2 + 𝐶𝐶
𝑥𝑥 2 − 10𝑥𝑥 =
What constant needs to be subtracted to compensate for the completed corner of the square?
𝑥𝑥 2 − 22𝑥𝑥 =
What constant needs to be subtracted to compensate for the completed corner of the square?
𝑥𝑥 2 − 𝑏𝑏𝑏𝑏 =
What constant needs to be subtracted to compensate for the completed corner of the square?
2) 𝑥𝑥 2 − 4𝑥𝑥 − 5
3) 𝑥𝑥 2 + 3𝑥𝑥 + 2
4) 2𝑥𝑥 2 + 5𝑥𝑥 − 3
5) 𝑥𝑥 2 + 2𝑥𝑥 − 9
6) 𝑥𝑥 2 − 7𝑥𝑥 − 1
7) 3𝑥𝑥 2 − 2𝑥𝑥 + 7
8) 4𝑥𝑥 2 − 4𝑥𝑥 − 3
1)
2)
3)
4)
2) 𝑥𝑥 − 4√𝑥𝑥 + 3 = 0
1 1
3) 𝑥𝑥 4 − 2𝑥𝑥 2 + 1 = 0
4) 3sin2 𝑥𝑥 + 2 sin 𝑥𝑥 − 1 = 0
5) 𝑥𝑥 4 = 10𝑥𝑥 2 − 9
4 3
6) + +1=0
𝑥𝑥 𝑥𝑥 2
7) cos 2 𝑥𝑥 − 2 cos 𝑥𝑥 = 8
8) 3 tan 𝑥𝑥 = 3 − tan2 𝑥𝑥
B D
Part A
Write down the correct inequalities in x which describe the sets of numbers identified on the
number line.
Part B
Solve the linear inequalities and write down the solution sets.
Each image consists of a parabola, a straight line and an identified region on the x-axis. What do
you notice about each image? Write down your ideas on this sheet.
First, think about the image each of the following quadratic inequalities would give i.e. what region
of the input-axis do you expect to see highlighted for the solution set.
Compare your final solution sets to the ones that you originally thought would occur.
(b) 𝑘𝑘 2 − 5𝑘𝑘 + 4 ≥ 0
(c) 𝑦𝑦(𝑦𝑦 − 1) ≤ 20
(e) 4𝑥𝑥 − 3 ≥ 𝑥𝑥 2
Worksheet A: Answers
Part 1:
1) 𝑥𝑥 2 + 6𝑥𝑥 + 9
2) 𝑥𝑥 2 + 20𝑥𝑥 + 100
3) 𝑥𝑥 2 + 16𝑥𝑥 + 64
4) 𝑥𝑥 2 − 16𝑥𝑥 + 64
Part 2:
1) The number hidden is 25. The dimensions of the square are 𝑥𝑥 + 5 and 𝑥𝑥 + 5.
2) The number hidden is 16. The dimensions of the square are 𝑥𝑥 + 4 and 𝑥𝑥 + 4.
3) The number hidden is 16. The dimensions of the square are 𝑥𝑥 − 4 and 𝑥𝑥 − 4.
Part 3:
1) A = -6, B = -31.
2) A = -1, B = -6, C = 41 .
3) A = 2, B = -3, C = -13.
4) A =4 , B = -3/2, C = 11/4.
Worksheet B: Answers
𝑏𝑏 2 𝑏𝑏 2 𝑏𝑏 2
3) We need to subtract , so that 𝑥𝑥 2 + 𝑏𝑏𝑏𝑏 = �𝑥𝑥 + � − .
4 2 4
Worksheet C: Answers
5 2 49
4) 2𝑥𝑥 2 + 5𝑥𝑥 − 3 = 2 �𝑥𝑥 + � −
4 8
2 2
5) 𝑥𝑥 + 2𝑥𝑥 − 9 = (𝑥𝑥 + 1) − 10
7 2 53
6) 𝑥𝑥 2 − 7𝑥𝑥 − 1 = �𝑥𝑥 − � −
2 4
1 2 20
7) 3𝑥𝑥 2 − 2𝑥𝑥 + 7 = 3 �𝑥𝑥 − � +
3 3
1 2
8) 4𝑥𝑥 2 − 4𝑥𝑥 − 3 = 4 �𝑥𝑥 − � − 4
2
1 2 20
7) 3𝑥𝑥 2 − 2𝑥𝑥 + 7 = 3 �𝑥𝑥 − � +
3 3
1 2
8) 4𝑥𝑥 2 − 4𝑥𝑥 − 3 = 4 �𝑥𝑥 − � − 4
2
Factorise and solve when equal to zero (for use with Lesson 2):
1) If 0 = 𝑥𝑥 2 + 2𝑥𝑥 + 1 = (𝑥𝑥 + 1)2 then 𝑥𝑥 = −1.
2) If 0 = 𝑥𝑥 2 − 4𝑥𝑥 − 5 = (𝑥𝑥 − 5)(𝑥𝑥 + 1) then 𝑥𝑥 = 5 or 𝑥𝑥 = −1.
3) If 0 = 𝑥𝑥 2 + 3𝑥𝑥 + 2 = (𝑥𝑥 + 2)(𝑥𝑥 + 1) then 𝑥𝑥 = −2 or 𝑥𝑥 = −1.
1
4) If 0 = 2𝑥𝑥 2 + 5𝑥𝑥 − 3 = (2𝑥𝑥 − 1)(𝑥𝑥 + 3) then 𝑥𝑥 = or 𝑥𝑥 = −3.
2
Complete the square and solve when equal to zero (for use with Lesson 2):
5) If 0 = 𝑥𝑥 2 + 2𝑥𝑥 − 9 = (𝑥𝑥 + 1)2 − 10 then 𝑥𝑥 = −1 ± √10.
7 2 53 7±√53
6) If 0 = 𝑥𝑥 2 − 7𝑥𝑥 − 1 = �𝑥𝑥 − � − then 𝑥𝑥 = .
2 4 2
1 2 20
7) If 0 = 3𝑥𝑥 2 − 2𝑥𝑥 + 7 = 3 �𝑥𝑥 − � + then no solutions.
3 3
1 2 3 1
8) If 0 = 4𝑥𝑥 2 − 4𝑥𝑥 − 3 = 4 �𝑥𝑥 − � − 4 then 𝑥𝑥 = or 𝑥𝑥 = − .
2 2 2
Worksheet D: Answers
Variation 1:
Variation 2:
Variation 3:
Worksheet E: Answers
1)
2)
3)
4)
Worksheet F: Answers
4) This is a quadratic in sin 𝑥𝑥. Let 𝑌𝑌 = sin 𝑥𝑥, then 3𝑌𝑌 2 + 2𝑌𝑌 − 1 = 0.
Worksheet G: Answers
A 𝑦𝑦 = 𝑥𝑥 2 − 4𝑥𝑥 − 5
B 𝑦𝑦 = −𝑥𝑥 2 + 4𝑥𝑥 + 5
C 𝑦𝑦 = 3𝑥𝑥 2 − 12𝑥𝑥 − 15
D 𝑦𝑦 = −2𝑥𝑥 2 + 8𝑥𝑥 + 10
Worksheet H: Answers
Part A:
A −5 < 𝑥𝑥 ≤ 10
B 𝑥𝑥 ≤ 2
C −1 < 𝑥𝑥
D 𝑥𝑥 = 1
Part B:
1) −1 < 𝑥𝑥
2) −5 < 𝑥𝑥 ≤ 10
3) 𝑥𝑥 ≤ 2
4) 𝑥𝑥 = 1
Worksheet J: Answers
a) 1 < 𝑥𝑥 < 4
b) 𝑘𝑘 < 1 or 𝑘𝑘 > 4
c) −4 ≤ 𝑦𝑦 ≤ 5
1
d) −2 < 𝑥𝑥 <
3
e) 1 ≤ 𝑥𝑥 ≤ 3
1 3
f) 𝑡𝑡 < − or 𝑡𝑡 >
2 5