Quadratic equations.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Quadratic equations

Equations and graphs

Lesson 3 of 6

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and graphs

Lesson 3 of 6

Answer each one, then check.

1

Factorise x² + 5x + 6.

2

Expand (x + 3)(x − 1).

3

What is √49?

4

If ab = 0, what do you know?

5

What is the value of 3²?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and graphs

Lesson 3 of 6

1

Factorise x² + 5x + 6.

(x + 2)(x + 3)

2

Expand (x + 3)(x − 1).

x² + 2x − 3

3

What is √49?

7

4

If ab = 0, what do you know?

a = 0 or b = 0

5

What is the value of 3²?

9

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and graphs

Lesson 3 of 6

1

Solve a quadratic by factorising.

2

Solve a quadratic with the quadratic formula.

3

Complete the square and use it to solve.

4

Choose the most efficient method.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and graphs

Lesson 3 of 6

QUADRATIC EQUATIONS

A quadratic equation can be written ax² + bx + c = 0 and usually has two solutions.

Rearrange so that one side is 0 before you use any method.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Two Solutions

Equations and graphs

Lesson 3 of 6

The solutions are where the graph crosses the x-axis.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Choosing a Method

Equations and graphs

Lesson 3 of 6

Use the quickest one that works.

Method

Use it when

Example

Factorising

The quadratic factorises easily

x² − 5x + 6 = 0

Formula

It does not factorise, or the answer is needed as a decimal

x² + 4x − 7 = 0

Completing the square

You need exact surds or the turning point

x² + 6x − 2 = 0

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Factorising

Equations and graphs

Lesson 3 of 6

Solve 2x² + 5x − 3 = 0.

1

Factorise

(2x − 1)(x + 3) = 0

2

Each bracket is 0

2x − 1 = 0 or x + 3 = 0

3

Solve

x = ½ or x = −3

ANSWER

x = ½ or x = −3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

The Quadratic Formula

Equations and graphs

Lesson 3 of 6

Solve x² + 4x − 7 = 0. Give your answers to 2 decimal places.

1

Formula

x = (−b ± √(b² − 4ac))/2a

2

Substitute a = 1, b = 4, c = −7

x = (−4 ± √(16 + 28))/2

3

Simplify

x = (−4 ± √44)/2

4

Evaluate

x = 1.32 or x = −5.32

ANSWER

x = 1.32 or x = −5.32

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Completing the Square

Equations and graphs

Lesson 3 of 6

Solve x² + 6x − 2 = 0 by completing the square. Give your answers to 2 decimal places.

1

Complete the square

(x + 3)² − 9 − 2 = 0

2

Simplify

(x + 3)² = 11

3

Square root

x + 3 = ± √11

4

Solve

x = −3 ± √11 = 0.32 or −6.32

ANSWER

x = 0.32 or x = −6.32

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Common Mistakes

Equations and graphs

Lesson 3 of 6

Avoid these.

Not rearranging to 0

Move everything to one side first.

Dividing by x

Solving x² = 7x by dividing loses the solution x = 0; factorise instead.

Signs in the formula

Take care with −b and b² − 4ac when b or c is negative.

Only one solution

Give both solutions.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and graphs

Lesson 3 of 6

Quadratic

An equation with a highest power of x².

Factorise

Write as a product of brackets.

Root

A solution of an equation.

Discriminant

b² − 4ac.

Completing the square

Writing x² + bx + c as (x + p)² + q.

Surd

A root that cannot be simplified to a whole number, e.g. √11.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and graphs

Lesson 3 of 6

YOUR TASK

Choose Your Method

12 minutes

Solve each and say which method you used. (a) x² − 7x + 10 = 0 (b) x² + 2x − 5 = 0 (2 d.p.) (c) x² = 9x.

1

Look for factors first.

2

Otherwise use the formula.

WHAT A GOOD ANSWER SHOWS

(a) x = 2 or 5 (factorising). (b) x = −1 ± √6 = 1.45 or −3.45 (formula or completing the square). (c) x(x − 9) = 0, so x = 0 or 9.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and graphs

Lesson 3 of 6

Rearrange to equal zero.

Factorise and solve.

Use the quadratic formula.

Complete the square.

Give answers to a given accuracy.

Give both solutions.

Avoid dividing by x.

Choose a method.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Factorise if you can.

Formula: x = (−b ± √(b² − 4ac))/2a.

Complete the square: (x + p)² + q.

Every quadratic has up to two solutions.

EXAM FOCUS

Solve x² + 4x − 7 = 0. Give your solutions correct to 2 decimal places. (3 marks)

Write the formula first, then substitute carefully, using brackets for negatives.

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