# Solving quadratic equations

This list contains resources to support the teaching and learning of how to solve quadratic equations and in so doing students will:

- identify and interpret roots, intercepts and turning points of quadratic functions graphically;
- deduce roots algebraically and turning points by completing the square

Visit the secondary mathematics webpage to access all lists.

### Quadratic Equations

This interactive resource, produced by the University of Leicester, is designed to enable students to explore what is meant by a quadratic equation, the meaning of the coefficients of a quadratic equation and to be able to solve quadratic equations.

An introduction page gives examples of where quadratic equations can be found which is useful for class discussion. There follows an explanation of the first task in which students have to investigate how changing the coefficients a, b and c in the function f(x) = a(x+b)2+c affect the graph.

The second activity requires students to change the values of a, b and c so that the green graph matches the blue graph on the screen.

The third activity asks students to form equations to match the paths shown on the screen.

The fourth activity explores the roots of a quadratic equation. This is followed by pages of notes explaining the nature of quadratic equations including the formula for solving quadratic equations, the determinant, factorising a quadratic and completing the square.

The final exercise asks students to solve a series of quadratic equations.

### Nuffield National Curriculum Mathematics 5

The Nuffield National Curriculum Mathematics stage five book is split into the same three sections. Each section is divided into chapters giving relevant information and presenting activities to do and questions to answer.

There are 13 chapters on **Number and algebra** at level eight and above, and are therefore suitable for those candidates following Higher GCSE courses. Each chapter is subdivided into sections showing the mathematics it covers. **Graphs of functions, algebraic manipulation and formuale, gradients** and **areas under a curve** are some of the topics tackled. The modern mathematics topic of **linear programming** is introduced in chapter seven.

Chapter 12 reviews solving quadratics including **completing the square **and asks the question does every quadratic have a solution? Chapter 13 looks at the **properties of quadratic functions** by using the completed square form to translate the graph of y= x2 . Students also consider the impact when using **function notation**.

### Quadratic Functions

This MEP resource from CIMT is taken from text book 9B.

Quadratic functions covers:

- a recap of drawing and transforming quadratic graphs,
- solving quadratics by factorisation
- solving quadratics by completing the square.
- The initial file forms part of the textbook. The activities sheet, extra exercises and mental tests compliment the work covered in the textbook. The overhead slides can be used on an interactive whiteboard.

Alongside the pupils' material there are lesson plans which outline the content of the unit, these are differentiated into two levels, A and E as well as suggested routes through them.

The material would form a good basic structure for a unit on solving quadratic equations. The lesson plans are just an outline but could be developed to include more detail. The material includes worked examples and a broad range of pupil examples including some applied examples.

Section 17.1 Reviews what students know about quadratics and looks at graphs of quadratic.

17.2 factorises basic quadratic algebraically.

17.3 introduces completing the square. There is a sound algebraic explanation but no use of visual images to support it. There are extra examples and 3 activities. These start to explore the origins of the formula and explore transformations and turning points.

### Graphs

**Graphical representations of quadratic equations (Exposing and discussing common misconceptions)**

This activityis part of the Mathematical Moments resources produced by the Learning and Skills Improvement Service (LSIS). The objectives is to explore the graphs of quadratic equations, specifically comparing those that factorise and those that do not. This allows review of prior learning about the solution of quadratic equations by factorisation and the need to be able to find a solution by other means. Students will also explore the idea that quadratic equations that do not factorise need to have another method of solution. This activity could be used as a follow-up to the classification of quadratics explored in ‘Mathematical Moments: Quadratic equations (Exposing and discussing common misconceptions). It could also be a useful introduction to the longer session C1 ‘Linking the properties and forms of quadratic equations’ which is also included in this list.

### Linking the Properties and Forms of Quadratic Equations C1

C1: Linking the properties and forms of quadratic equations

In this DfE Standards Unit resource students identify different forms and properties of quadratic functions, connect quadratic functions with their graphs and properties, including intersections with axes, maxima and minima. Students will need to be familiar with the

following forms of quadratic functions:

y = ax2+ bx + c,

y = (x + a)(x + b),

y = a(x + b)2+ c

Learners are asked to produce a graph from a given quadratic function, with intercepts and stationary points marked. Different levels of challenge could be available for students to choose from. There is the opportunity for completed square form to be linked to translations of the graphs.

The activity uses card sorts and discussion followed by written justification of examples to consolidate and extend students existing knowledge.

These materials exemplify the ideas and approaches adopted in the Standards Unit pilot. To get best value from them use them in association with the guidance and other materials in Improving Learning in Mathematics: Challenges and Strategies.

### Quadratics

This resource from The Virtual Texbook contains four excel programs with interactive spreadsheets dealing with topics relating to quadratic expressions, their graphs and solving quadratic equations. Topics covered include:

- Rearranging and graphing quadratics
- Solving quadratics
- Tables of values for quadratics
- Turning point