Completing the square.pptx

EDEXCEL GCSE MATHS · HIGHER

Completing the square

Equations and inequalities

Lesson 4 of 7

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Warm-up

Equations and inequalities

Lesson 4 of 7

Answer each one, then check.

1

Expand (x + 3)².

2

Expand (x − 4)².

3

Work out √11 to 2 decimal places.

4

Half of 6 is what?

5

What is the turning point of y = x²?

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Warm-up - Answers

Equations and inequalities

Lesson 4 of 7

1

Expand (x + 3)².

x² + 6x + 9

2

Expand (x − 4)².

x² − 8x + 16

3

Work out √11 to 2 decimal places.

3.32

4

Half of 6 is what?

3

5

What is the turning point of y = x²?

(0, 0)

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Learning Objectives

Equations and inequalities

Lesson 4 of 7

1

Write x² + bx + c in the form (x + p)² + q.

2

Complete the square when the coefficient of x² is not 1.

3

Solve quadratic equations by completing the square.

4

Find the turning point of a quadratic graph.

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Equations and inequalities

Lesson 4 of 7

THE METHOD

x² + bx + c = (x + b/2)² − (b/2)² + c

Halve the coefficient of x, square it, and subtract it.

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Completing the Square

Equations and inequalities

Lesson 4 of 7

Write x² + 6x + 5 in the form (x + p)² + q.

1

Halve the coefficient of x

6 ÷ 2 = 3, so (x + 3)²

2

(x + 3)² expands to

x² + 6x + 9

3

Adjust the constant: we need 5, not 9

5 − 9 = −4

4

Write the result

(x + 3)² − 4

ANSWER

(x + 3)² − 4

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

A Negative Coefficient

Equations and inequalities

Lesson 4 of 7

Write x² − 8x + 3 in the form (x + p)² + q.

1

Halve −8

−4, so (x − 4)²

2

(x − 4)² expands to

x² − 8x + 16

3

Adjust the constant

3 − 16 = −13

ANSWER

(x − 4)² − 13

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

What the Form Tells You

Equations and inequalities

Lesson 4 of 7

In y = (x + p)² + q the turning point is (−p, q).

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Reading the Completed Square Form

Equations and inequalities

Lesson 4 of 7

y = (x + 3)² − 4

Turning point (−3, −4): the sign inside the bracket is reversed.

Line of symmetry

x = −3.

Minimum value

−4, when x = −3.

Never below the minimum

(x + 3)² ≥ 0, so y ≥ −4.

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Solving by Completing the Square

Equations and inequalities

Lesson 4 of 7

Solve x² + 6x − 2 = 0, leaving your answer in surd form.

1

Complete the square

(x + 3)² − 9 − 2 = 0, so (x + 3)² − 11 = 0

2

Rearrange

(x + 3)² = 11

3

Take the square root of both sides

x + 3 = ± √11

4

Solve

x = −3 ± √11

ANSWER

x = −3 + √11 or x = −3 − √11

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

A Coefficient of x² Not 1

Equations and inequalities

Lesson 4 of 7

Write 2x² + 12x + 7 in the form a(x + p)² + q.

1

Take out the 2 from the x terms

2(x² + 6x) + 7

2

Complete the square inside

2[(x + 3)² − 9] + 7

3

Multiply out the 2

2(x + 3)² − 18 + 7

4

Simplify

2(x + 3)² − 11

ANSWER

2(x + 3)² − 11

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

A Proof

Equations and inequalities

Lesson 4 of 7

Show that x² + 4x + 5 is always positive.

1

Complete the square

x² + 4x + 5 = (x + 2)² + 1

2

A square is never negative

(x + 2)² ≥ 0

3

Add 1

(x + 2)² + 1 ≥ 1

ANSWER

(x + 2)² + 1 ≥ 1 > 0, so the expression is always positive.

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Key Terms

Equations and inequalities

Lesson 4 of 7

Completing the square

Rewriting a quadratic as a squared bracket plus a constant.

Turning point

The lowest or highest point of a quadratic graph.

Minimum value

The smallest value a function can take.

Line of symmetry

The vertical line through the turning point.

Surd

A root left in exact form, such as √11.

Coefficient

The number multiplying a variable.

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Equations and inequalities

Lesson 4 of 7

YOUR TASK

Complete the Square Race

12 minutes

Write each in the form (x + p)² + q, then state the turning point of its graph. (a) x² + 4x + 1 (b) x² − 10x + 30 (c) x² + 2x − 8 (d) x² − 3x.

1

Halve the coefficient of x.

2

Subtract its square.

3

Read off the turning point.

WHAT A GOOD ANSWER SHOWS

(a) (x + 2)² − 3, turning point (−2, −3). (b) (x − 5)² + 5, turning point (5, 5). (c) (x + 1)² − 9, turning point (−1, −9). (d) (x − 3/2)² − 9/4, turning point (3/2, −9/4).

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Can I...?

Equations and inequalities

Lesson 4 of 7

Complete the square when a = 1.

Complete the square when a ≠ 1.

Find the turning point from the completed form.

Solve by completing the square.

Give answers as surds.

Use the form to find the minimum value.

Prove a quadratic is always positive.

Check by expanding.

openrevise.com

EDEXCEL GCSE MATHS · HIGHER

Summary & Exam Focus

x² + bx + c = (x + b/2)² + c − (b/2)².

Turning point of y = (x + p)² + q is (−p, q).

Solve by rearranging to (x + p)² = k.

(x + p)² ≥ 0 helps with proofs.

EXAM FOCUS

Write x² − 8x + 3 in the form (x + a)² + b. (2 marks)

Halve the coefficient of x (keeping its sign) and square it. Then adjust the constant. Always check by expanding.

openrevise.com