Solving quadratic equations 1.pptx

EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving quadratic equations 1

Equations and inequalities

Lesson 2 of 7

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up

Equations and inequalities

Lesson 2 of 7

Answer each one, then check.

1

Factorise x² + 5x + 6.

2

Factorise x² − 9.

3

Solve x − 4 = 0.

4

Expand (x + 2)(x + 3).

5

What is the product of 0 and any number?

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Warm-up - Answers

Equations and inequalities

Lesson 2 of 7

1

Factorise x² + 5x + 6.

(x + 2)(x + 3)

2

Factorise x² − 9.

(x − 3)(x + 3)

3

Solve x − 4 = 0.

x = 4

4

Expand (x + 2)(x + 3).

x² + 5x + 6

5

What is the product of 0 and any number?

0

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Learning Objectives

Equations and inequalities

Lesson 2 of 7

1

Solve x² + bx + c = 0 by factorising.

2

Solve equations of the form x² − a² = 0.

3

Rearrange to zero before factorising.

4

Solve problems that lead to quadratic equations.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 2 of 7

THE NULL FACTOR LAW

If two things multiply to give zero, at least one of them must be zero.

So (x − 2)(x − 3) = 0 means x = 2 or x = 3.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Roots on a Graph

Equations and inequalities

Lesson 2 of 7

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

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Solving by Factorising

Equations and inequalities

Lesson 2 of 7

Always get zero on one side first.

1

Rearrange

Write the equation as ax² + bx + c = 0

2

Factorise

Find two numbers that multiply to c and add to b

3

Set each bracket to zero

Use the null factor law

4

Solve

Two solutions, or one repeated solution

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Standard Quadratic

Equations and inequalities

Lesson 2 of 7

Solve x² + 5x + 6 = 0.

1

Two numbers with product 6 and sum 5

2 and 3

2

Factorise

(x + 2)(x + 3) = 0

3

Set each bracket to zero

x + 2 = 0 or x + 3 = 0

ANSWER

x = −2 or x = −3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

With Negative Numbers

Equations and inequalities

Lesson 2 of 7

Solve x² − 2x − 15 = 0.

1

Product −15, sum −2

−5 and 3

2

Factorise

(x − 5)(x + 3) = 0

3

Solve

x = 5 or x = −3

ANSWER

x = 5 or x = −3

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Special Cases

Equations and inequalities

Lesson 2 of 7

Difference of two squares

x² − 49 = 0 factorises as (x − 7)(x + 7), so x = 7 or x = −7.

No constant term

x² − 7x = 0 factorises as x(x − 7), so x = 0 or x = 7.

A perfect square

x² − 6x + 9 = 0 is (x − 3)², so there is one solution, x = 3.

Do not divide by x

Dividing x² = 7x by x loses the solution x = 0.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Rearranging First

Equations and inequalities

Lesson 2 of 7

Solve x² = 3x + 10.

1

Move everything to one side

x² − 3x − 10 = 0

2

Factorise

(x − 5)(x + 2) = 0

3

Solve

x = 5 or x = −2

ANSWER

x = 5 or x = −2

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

A Rectangle Problem

Equations and inequalities

Lesson 2 of 7

A rectangle has length (x + 3) cm and width x cm. Its area is 40 cm². Find x, and the dimensions.

1

Area equation

x(x + 3) = 40

2

Expand and rearrange

x² + 3x − 40 = 0

3

Factorise

(x + 8)(x − 5) = 0

4

A length cannot be negative

x = 5, and x = −8 is rejected

ANSWER

x = 5; the rectangle is 8 cm by 5 cm.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Key Terms

Equations and inequalities

Lesson 2 of 7

Quadratic equation

An equation whose highest power is x².

Root

A solution of an equation; where a graph crosses the x-axis.

Factorise

Write as a product of brackets.

Null factor law

If ab = 0, then a = 0 or b = 0.

Difference of two squares

a² − b² = (a − b)(a + b).

Perfect square

A quadratic that is a bracket squared.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Equations and inequalities

Lesson 2 of 7

YOUR TASK

Solve and Check

12 minutes

Solve each equation by factorising, then substitute your answers back to check. (a) x² − 8x + 15 = 0 (b) x² + 4x = 0 (c) x² − 25 = 0 (d) x² = 2x + 24.

1

Rearrange to zero.

2

Factorise.

3

Check by substituting.

WHAT A GOOD ANSWER SHOWS

(a) (x − 3)(x − 5) = 0: x = 3 or 5. (b) x(x + 4) = 0: x = 0 or −4. (c) (x − 5)(x + 5) = 0: x = ± 5. (d) x² − 2x − 24 = 0, (x − 6)(x + 4) = 0: x = 6 or −4.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Can I...?

Equations and inequalities

Lesson 2 of 7

Factorise x² + bx + c.

Use the null factor law.

Solve a difference of two squares.

Solve an equation with no constant term.

Rearrange to zero first.

Reject impossible solutions.

Form a quadratic from a problem.

Check by substituting.

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EDEXCEL GCSE MATHS · FOUNDATION & HIGHER

Summary & Exam Focus

Make the equation equal zero before factorising.

Set each bracket to zero.

Difference of two squares: x² − a² = (x − a)(x + a).

Check solutions make sense in context.

EXAM FOCUS

A rectangle has length (x + 3) cm and width x cm. The area of the rectangle is 40 cm². Work out the value of x. (4 marks)

Form the equation, rearrange to zero, factorise, and reject any negative length.

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