Topic Two (1.2): Quadratics

Introduction

A quadratic function contains at least one term that is squared. The standard form is LaTeX: ax^2+bx+c=0 $ax^2+bx+c=0$ with $LaTeX: a,\:b$ $a,\:b$ and LaTeX: c $c$ being constants or numerical coefficients (where $LaTeX: a\ne0$ $a\ne0$ ) and LaTeX: x $x$ is an unknown variable. Quadratics can be solved in several ways, each of which requires setting the quadratic to LaTeX: 0 $0$ first.

Factorising is simplest method, providing the quadratic can be factorised.

Using the quadratic formula: $LaTeX: x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ . If a quadratic cannot be factorised but does have roots, then the quadratic formula will find them. The discriminant is the part of the quadratic formula underneath the square root symbol: LaTeX: b^2-4ac $b^2-4ac$ . It tells us how many roots the quadratic has.

Completing the square is used to derive the quadratic formula. It is a useful tool for sketching the quadratic, because when the quadratic is written in completed square form, it is possible to see the transformations that are applied to the graph of LaTeX: x^2 $x^2$ .

Unlike linear functions, quadratic functions have maximum and minimum values and their graphs are symmetrical.

In the early 17^th century Galileo discovered that the trajectory of a ball as it travels through the air is a quadratic. The vertical motion of a ball thrown straight upwards can also be modelled by a quadratic.

Reading

For this topic, you will need to work through pages 1 to 32 of your textbook.

Tasks

Complete the following:

Start by reading through pages 1 to 2 as an introduction to this chapter.

Solving quadratic equations by factorisation

Work through pages 3 to 6, including exercise 1A.
You may wish to watch the following videos:

WATCH & EXPLORE – Factorising Expressions (Khan Academy) Links to an external site.

WATCH & EXPLORE – Solving Quadratic Equations by Factoring (Khan Academy) Links to an external site.

WATCH & EXPLORE – Solving Quadratics by Factoring (Khan Academy) Links to an external site.

Completing the square

Work through pages 6 to 9, including exercise 1B.
This video explains completing the square, should you need extra help with this:

WATCH – Completing the Square (Exam Solutions) Links to an external site.

The quadratic formula

Work through pages 10 to 11, including exercise 1C.
This video contains worked examples:

WATCH – Solve by the Quadratic Formula (Exam Solutions) Links to an external site.

Solving simultaneous equations (one linear and one quadratic)

Work through pages 11 to 14, including exercise 1D.

WATCH – Substitution Method for Linear and Non-linear Equations (Exam Solutions) Links to an external site.

Solving more complex quadratic equations

Work through pages 15 to 17, including exercise 1E.

Maximum and minimum values of a quadratic function

Work through pages 17 to 21, including exercise 1F.

Solving quadratic inequalities

Work through pages 21 to 24, including exercise 1G.
There are two ways of solving quadratic inequalities; most students prefer sketching the curve (as explained in the textbook) and in the following tutorial:

WATCH – Quadratic Inequalities (Exam Solutions) Links to an external site.

However, an alternative method is using a sign chart, as explained in:

WATCH – How to Solve a Quadratic Inequality using a Sign Chart (YouTube) Links to an external site.

The number of roots of a quadratic equation

Work through pages 24 to 26, including exercise 1H.

Intersection of a line and a quadratic curve

Work through pages 27 to 29, including exercise 1I.

Consolidate

Now consolidate your understanding by making revision cards for this chapter (the checklist on page 30 will help) and work through the end-of-chapter review on pages 31 to 32.

Assignment

When you have completed all the activities and are fully prepared and feel confident with the material, you should complete Assignment One and submit it to your tutor via Canvas for marking and feedback.