Solving quadratic equations 2.pptx

EDEXCEL GCSE MATHS · HIGHER

Solving quadratic equations 2

Equations and inequalities

Lesson 3 of 7

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EDEXCEL GCSE MATHS · HIGHER

Warm-up

Equations and inequalities

Lesson 3 of 7

Answer each one, then check.

1

Solve x² − 5x + 6 = 0.

2

Expand (2x + 1)(x + 3).

3

Work out √81.

4

Work out 5² − 4 × 1 × 6.

5

Round 5.385 to 2 decimal places.

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EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

Equations and inequalities

Lesson 3 of 7

1

Solve x² − 5x + 6 = 0.

x = 2 or x = 3

2

Expand (2x + 1)(x + 3).

2x² + 7x + 3

3

Work out √81.

9

4

Work out 5² − 4 × 1 × 6.

1

5

Round 5.385 to 2 decimal places.

5.39

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EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

Equations and inequalities

Lesson 3 of 7

1

Factorise and solve ax² + bx + c = 0.

2

Use the quadratic formula.

3

Give answers to a given accuracy or as surds.

4

Decide which method to use.

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EDEXCEL GCSE MATHS · HIGHER

Factorising ax² + bx + c

Equations and inequalities

Lesson 3 of 7

Use the product ac and the sum b.

1

Multiply a by c

For 2x² + 7x + 3: 2 × 3 = 6

2

Find two numbers with that product and sum b

1 and 6 (they add to 7)

3

Split the middle term

2x² + x + 6x + 3

4

Factorise in pairs

x(2x + 1) + 3(2x + 1)

5

Write as two brackets

(2x + 1)(x + 3)

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EDEXCEL GCSE MATHS · HIGHER

A Harder Factorisation

Equations and inequalities

Lesson 3 of 7

Solve 2x² + 7x + 3 = 0.

1

Factorise

(2x + 1)(x + 3) = 0

2

Set each bracket to zero

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 · HIGHER

Another Factorisation

Equations and inequalities

Lesson 3 of 7

Solve 3x² − 10x + 8 = 0.

1

ac = 24; two numbers with product 24 and sum −10

−4 and −6

2

Split and factorise

3x² − 4x − 6x + 8 = x(3x − 4) − 2(3x − 4)

3

Write as brackets

(3x − 4)(x − 2) = 0

4

Solve

x = 4/3 or x = 2

ANSWER

x = 4/3 or x = 2

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EDEXCEL GCSE MATHS · HIGHER

Equations and inequalities

Lesson 3 of 7

THE QUADRATIC FORMULA

For ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac))/2a.

Learn it: it is not on the formulae sheet.

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EDEXCEL GCSE MATHS · HIGHER

When It Will Not Factorise

Equations and inequalities

Lesson 3 of 7

The roots are not whole numbers, so factorising will not work.

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EDEXCEL GCSE MATHS · HIGHER

Using the Formula

Equations and inequalities

Lesson 3 of 7

Solve x² + 3x − 5 = 0, giving your answers to 2 decimal places.

1

Identify a = 1, b = 3, c = −5

Substitute into the formula

2

b² − 4ac

9 − 4(1)(−5) = 9 + 20 = 29

3

Write the formula

x = (−3 ± √29)/2

4

Calculate

√29 = 5.385...

5

The two solutions

x = 1.19 or x = −4.19

ANSWER

x = 1.19 or x = −4.19

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EDEXCEL GCSE MATHS · HIGHER

Leaving the Answer as a Surd

Equations and inequalities

Lesson 3 of 7

Solve x² − 6x + 2 = 0, giving your answers in the form p ± √q.

1

a = 1, b = −6, c = 2

b² − 4ac = 36 − 8 = 28

2

Formula

x = (6 ± √28)/2

3

Simplify √28 = 2√7

x = (6 ± 2√7)/2

4

Divide through by 2

x = 3 ± √7

ANSWER

x = 3 + √7 or x = 3 − √7

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EDEXCEL GCSE MATHS · HIGHER

Which Method?

Equations and inequalities

Lesson 3 of 7

FACTORISING

• Quick when the numbers are simple.

• Works when the roots are rational.

• Gives exact answers.

THE QUADRATIC FORMULA

• Always works.

• Needed for "give to 2 decimal places" or surds.

• Take care with the signs of b and c.

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EDEXCEL GCSE MATHS · HIGHER

Common Mistakes

Equations and inequalities

Lesson 3 of 7

The formula has plenty of places to slip.

Sign of b

The formula starts with −b; if b = −6, then −b = 6.

The bracket

b² must be squared before any minus: (−6)² = 36.

Dividing

Divide the WHOLE numerator by 2a, not just the surd.

Order of the calculation

Use brackets on the calculator for the top line.

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EDEXCEL GCSE MATHS · HIGHER

Key Terms

Equations and inequalities

Lesson 3 of 7

Coefficient

The number multiplying a variable, such as the 2 in 2x².

Quadratic formula

x = (−b ± √(b² − 4ac))/2a.

Discriminant

The expression b² − 4ac under the square root.

Surd

A root that cannot be simplified to a whole number, such as √7.

Root

A solution of an equation.

Substitute

Replace letters with numbers.

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EDEXCEL GCSE MATHS · HIGHER

Equations and inequalities

Lesson 3 of 7

YOUR TASK

Choose Your Method

12 minutes

For each equation, decide whether to factorise or use the formula, then solve it. (a) 2x² − 5x − 12 = 0 (b) x² + 4x − 3 = 0 (to 2 d.p.) (c) x² − 10x + 25 = 0.

1

Test for simple factors.

2

Otherwise use the formula.

WHAT A GOOD ANSWER SHOWS

(a) (2x + 3)(x − 4) = 0: x = −1.5 or 4. (b) Formula: (−4 ± √28)/2 = −2 ± √7: x = 0.65 or −4.65. (c) (x − 5)² = 0: x = 5 (one repeated solution).

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EDEXCEL GCSE MATHS · HIGHER

Can I...?

Equations and inequalities

Lesson 3 of 7

Factorise ax² + bx + c.

Solve by factorising.

Write down the quadratic formula.

Identify a, b and c correctly.

Substitute carefully with brackets.

Give answers to 2 decimal places.

Give answers as surds.

Choose between methods.

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EDEXCEL GCSE MATHS · HIGHER

Summary & Exam Focus

x = (−b ± √(b² − 4ac))/2a.

Factorise when the numbers are simple; otherwise use the formula.

Watch the signs and divide the whole numerator by 2a.

Check by substituting your answers into the original equation.

EXAM FOCUS

Solve x² + 3x − 5 = 0. Give your solutions correct to 2 decimal places. (3 marks)

Write a, b and c first, and put brackets round negative numbers when you substitute.

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