quadratic equations · 2017-07-12 · 1 factorise a 3x 1 6y b 8x 1 x2 c 3m2 1 mn ... b what do you...

20

Before you start this chapterPut your calculator away!

1 Factorisea 3x 1 6y b 8x 1 x2 c 3m2 1 mnd 5r2 1 15rt e 12xyz 1 6xy 1 18y2

2 Work outa 23 3 4 b 8 3 26 3 21 c 27 3 5 1 12

d (2 1 24) 3 23 e (26)2 3 3 f 7 3 24 1 4 ___________ 212

3 Work out the value of these expressions when a 5 23, b 5 24 and c 5 2.a abc b √

____ 24b

c ab2 2 c d b2 2 4ac

Quadratic equationsObjectives

This chapter will show you how to• factorise quadratic expressions, including the difference of two squares B A

• solve quadratic equations by rearranging B

• factorise quadratics and solve quadratic equations of the form ax2 1 bx 1 c 5 0 B A A*

• use the quadratic equations formula A A*

• complete the square A*

This chapter is about quadratics.

In nature, the growth of a population of rabbits can be modelled by a quadratic equation.

HELP Chapter 12

HELP Chapter 15

298 Quadratic equations

Exercise 20A1 Factorise

a x2 2 16 b x2 2 25 c x2 2 100 d x2 2 144e a2 2 1 f m2 2 64 g n2 2 36 h t2 2 121

2 Joe thinks of a number, squares it and subtracts 16.

a Write down an algebraic expression to illustrate this.

b Factorise your answer to part a.

B

AO2

B

L20.1 Factorising the difference

of two squares

Why learn this?Being able to factorise

a quadratic expression will help when solving

quadratic equations.

Skills check1 Expand

a (x 1 3)(x 2 3) b (x 1 7)(x 2 7) c (x 2 5)(x 1 5)What do you notice when you have expanded the brackets?

2 Expanda (2a 1 4)(2a 2 4) b (3x 2 2)(3x 1 2) c (5m 1 n)(5m 2 n)

ObjectivesB A Factorise a quadratic expression

that is the difference of two squares

Quadratic expressionsA quadratic expression is an algebraic expression whose highest power of x is x2.

They are usually of the form ax2 1 bx 1 c, where a, b and c are numbers and a 0.

These are all quadratic expressions.

3x2 1 2x 1 5 x2 2 3x 2 2 4x2 1 7 12x2 2 3x

These expressions all represent one square number subtracted from another.

x2 2 4 c2 2 64 16a2 2 25

An expression of the form x2 2 b2, where x and b are numbers or algebraic terms, is called the difference of two squares.In general, x2 2 b2 5 (x 2 b)(x 1 b). Check this by multiplying out (x 2 b)(x 1 b).

Keywordsquadratic expression, difference of two squares, factorise

HELP Section 12.5

Factorise x2 2 9.

Example 1

x2 2 9 5 x2 2 32

5 (x 2 3)(x 1 3)Write as ‘letter squared’ 2 ‘number squared’.

Use x2 2 b2 5 (x 2 b)(x 1 b) with b 5 3.

Remember that factorising is the inverse of expanding brackets.B

29920.2 Factorising quadratics of the form x2 1 bx 1 c


ObjectivesB Factorise a quadratic expression

of the form x2 1 bx 1 c

Why learn this?Understanding how an algebraic

expression is constructed can tell you much more about the

expression.

Skills check1 Find two positive numbers whose

a product is 12 and sum is 7 b product is 20 and sum is 12c product is 12 and sum is 13 d product is 212 and sum is 1e product is 212 and sum is 21 f product is 212 and sum is 24.

L

Factorising quadratics of the form x2 1 bx 1 cExpanding a product of two expressions, like (x 1 2) and (x 1 3), gives a quadratic expression.

(x 1 2)(x 1 3) 5 x2 1 5x 1 6

Keywordsproduct, sum, coefficient

Exercise 20B1 Factorise

a 4x2 2 25 b 9a2 2 36 c 16m2 2 1

d 100t2 2 121 e 169z2 2 4 f 225q2 2 144

2 Copy and complete. Take out 2 as a common factor.

72m2 2 50 5 2( 2 )

5 2( 2 )( 1 )

3 Factorise each expression by first taking out a common factor.

a 50x2 2 200 b 27m2 2 3 c 80t2 2 45

d 2a2 2 18b2 e 3h2 2 75k2 f 600x2 2 6y2

A

Factorise 16m2 2 49.

Example 2

16m2 2 49 5 (4m)2 2 72

5 (4m 2 7)(4m 1 7)

Notice that 16m2 5 (4m)2.

A


Exercise 20C1 Factorise each quadratic expression.

a x2 1 5x 1 6 b x2 1 6x 1 8 c z2 1 6z 1 5

d a2 1 11a 1 10 e n2 1 8n 1 15 f f 2 1 12f 1 36

g m2 1 8m 1 12 h x2 1 14x 1 24 i b2 1 11b 1 30

Factorising is the inverse of expanding.

To factorise a quadratic, you need to write it as the product of two expressions.

5 is the sum of 2 and 3.

x2 1 5x 1 6 5 x2 1 2x 1 3x 1 6 5 (x 1 2)(x 1 3)

6 is the product of 2 and 3.

In general, to factorise the equation x2 1 bx 1 c, find two The coefficient of x is the number multiplying the x.

numbers whose sum is b (the coefficient of x) and whose product is c.

Exercise 20D1 Factorise each quadratic expression.

a x2 2 5x 1 6 b x2 2 9x 1 8 c z2 2 7z 1 12

d a2 2 9a 1 18 e n2 2 10n 1 25 f f 2 2 8f 1 16

g x2 2 13x 1 30 h b2 2 11b 1 28 i p2 2 10p 1 24

B

B

Factorise x2 2 7x 1 10.

Example 4

x2 2 7x 1 10 5 (x 2 2)(x 2 5)

Find two numbers whose product is 10 and whose sum is 27.The pairs of numbers whose product is 10 are

1 and 10 1 1 10 5 11 72 and 5 2 1 5 5 7 721 and 210 21 1 210 5 211 722 and 25 22 1 25 5 27 3

The numbers must be 22 and 25.

B

B Factorise x2 1 7x 1 10.

Example 3

x2 1 7x 1 10 5 (x 1 5)(x 1 2)


1 and 10 1 1 10 5 11 72 and 5 2 1 5 5 7 3

The numbers must be 2 and 5.First look for the product, then test the sums.


Exercise 20E1 Factorise

a x2 1 4x 2 12 b x2 2 x 2 20 c z2 2 2z 2 15

d a2 1 6a 2 7 e n2 1 6n 2 16 f f 2 2 f 2 30

g m2 1 m 2 30 h t2 2 6t 2 72 i y2 1 19y 2 120

2 Copy and complete these statements.

a t2 1 7r 2 5 (t 1 10)(t 2 )

b m2 2 1 15 5 (t )(t 2 5)

c q2 2 12q 5 (q )(q 2 2)

3 a Factorise each expression. Simplify your answers as much as possible. i x2 1 6x 1 9 ii x2 2 8x 1 16 iii x2 1 4x 1 4 iv x2 2 14x 1 49 v x2 2 10x 1 25 vi x2 1 16x 1 64

b What do you notice about all the answers to part a?

c Copy and complete these statements, where m and n are numbers. i (x 1 m)2 5 x2 1 x 1

ii (x 2 n)2 5 x2 2 x 1

General rules for factorising quadraticsIn general

• if c is positive and b is positive, both numbers in the brackets will be positive

• if c is positive and b is negative, both numbers in the brackets will be negative

• if c is negative, one number will be negative, one will be positive.

B

BFactorise a x2 2 6x 2 7 b x2 1 x 2 12

Example 5

a x2 2 6x 2 7 5 (x 2 7)(x 1 1) b x2 1 x 2 12 5 (x 2 3)(x 1 4)


21 and 7 21 1 7 5 6 71 and 27 1 1 27 5 26 3

The numbers must be 1 and 27.


21 and 12 21 1 12 5 11 71 and 212 1 1 212 5 211 722 and 6 22 1 6 5 4 72 and 26 2 1 26 5 24 723 and 4 23 1 4 5 1 33 and 24 3 1 24 5 21 7

The numbers must be 23 and 4.With practice, you will become better at spotting the correct combination.


Exercise 20F1 Solve these equations.

a r2 5 169 b x2 1 5 5 14 c 18 5 2t2

d y2 2 20 5 219 e m2 ___

4 5 6.25 f

p2

__ 5 5 20

2 Find the roots of

a 2x2 1 7 5 39 b 5y2 2 100 5 280 Root is another name for a solution.c 3r2 5 5r2 2 98 d 3t2 5 t2 1 18

e 100 2 5y2 5 95 f 2x2 1 2 5 130

B

20.3 Solving quadratic equations

Why learn this?The path of a cricket

ball can be modelled using a quadratic

equation.

Skills check1 Solve the equation 2x 1 5 510x 2 19.2 Angel has x CDs in her collection. Write an algebraic expression

for the number of CDs that each of these friends has.a Amy who has twice as many as Angel. b Judith who has four less than Amy.c Angela who has half as many as Judith. d Jo who has four times as many as Angela.

ObjectivesB Solve quadratic equations by rearrangingB Solve quadratic equations by factorising

Keywordssolve, square root, root

Solving quadratic equations by rearrangingYou can solve some quadratic equations by rearranging them to make x the subject.

HELP Section 14.2

HELP Section 15.1

Solve the quadratic equation 3x2 2 27 5 0.

Example 6

3x2 2 27 5 0

3x2 5 27

x2 5 9

x 5 63

Add 27 to both sides of the equation.

Divide both sides by 3.

Find the square root of both sides. Remember, when you find the root there are two solutions: positive and negative.

B

30320.3 Solving quadratic equations

Solve the equation 2(x 1 3)2 2 5 5 195.

Example 7

2(x 1 3)2 2 5 5 195

2(x 1 3)2 5 200

(x 1 3)2 5 100

x 1 3 5 610

x 5 213 or 7

Add 5 to both sides of the equation.

Divide both sides by 2.

Take the square root of both sides.

Subtract 3 from both sides. Remember to give both solutions.

B

Exercise 20G1 Find the roots of

a 2(x 1 1)2 5 8 b 4.5 5 (r 2 7)2

_______ 2

2 Solve these equations.

a (x 1 1)2 2 16 5 20 b 100 5 4t2 1 36 c 150 2 3t2 5 42

d 6 1 3t2 5 2t2 1 15 e 4(x 1 3)2 5 100 f 7(y 2 2)2 5 700

3 A field is three times as long as it is wide.

a Using x for the width of the field, write an expression for its length.

b Write an expression for the area Use your answer to part a. of the field, in terms of x.

c The field has an area of 1200 m2. Write an equation for the area of the field. Use your answer to part b.

d Solve your equation to find x.

e What are the length and the width of the field?

4 Explain why you cannot find a solution to x2 1 20 5 5.

5 A rectangle has length five times its width. The area of the rectangle is 845 mm2. What is the width of the rectangle?

B

AO3

B

Solving quadratic equations by factorisingSolving quadratic equations by factorising relies on the fact that when a 3 b 5 0, a is 0, b is 0 or both are 0.

So if (x 1 2)(x 2 4) 5 0, either x 1 2 5 0, which If the product of two things is zero, one of them must be zero.

means x 5 22, or x 2 4 5 0, which means x 5 4.

To solve a quadratic equationStep 1: Rearrange the equation so that one side is zero.

Step 2: Factorise the quadratic expression.

Step 3: Find the solutions.

Usually there are two solutions. However, when the expression factorises to (x 1 m)2 5 0, there is only one solution.

AO2

B


B

Exercise 20H1 Solve these equations.

Factorise first.

a x2 1 7x 5 0 b t2 2 5t 5 0 c 3x2 1 6x 5 0

d y2 5 5y e 0 5 4w2 2 12w f 5y 5 20y2

g a 2 a2 5 0 h 5t 5 30t2 i 14r 5 63r2

2 Solve these equations.

a 2x2 2 8x 5 0 b 4t2 1 t 5 0 c 7m2 5 14m

d 8g2 5 24g e 15f 5 6f 2 f 35w 5 10w2

B

Solve the equation x2 5 3x.

Example 8

Step 1: x2 2 3x 5 0

Step 2: x(x 2 3) 5 0

Step 3: x 5 0 or x 2 3 5 0

So x 5 0 or x 5 3

Subtract 3x from both sides to make one side zero.

Factorise the expression.

Solve the equation. If the product of two numbers is zero, at least one of the numbers must be zero.

B

Find the roots of the equation x2 2 x 2 8 5 4.

Example 9

Step 1: x2 2 x 2 12 5 0

Step 2: (x 2 4)(x 1 3) 5 0

Step 3: x 2 4 5 0 or x 1 3 5 0

So x 5 4 or x 5 23

Subtract 4 from both sides to make one side zero.

Factorise the expression.

BRoot is another name for a solution.

Exercise 20I1 Find the roots of these equations.

a x2 1 4x 1 3 5 0 b x2 2 x 2 6 5 0 c x2 2 6x 1 8 5 0

d x2 1 x 5 12 e x2 5 x 1 20 f x2 1 2x 5 21

g z2 5 3z 1 4 h 2q 1 q2 5 15 i w2 5 4w 2 4

j 6t 1 7 5 t2 k 6p 1 9 5 2p2 l 10x 2 25 5 x2

2 Jane is three years younger than her older sister. The product of their ages is 54. Use x to represent Jane’s age.

a Write down an algebraic expression for her sister’s age.

Remember that Jane is younger.

b Write down and simplify an algebraic expression for the product of their ages.

c Form and solve an algebraic equation to find the value of x.

d Explain why only one of the solutions makes sense.AO2

B

30520.4 Factorising quadratics of the form ax2 1 bx 1 c

3 The height of the rectangle is 3 cm more than the width.

t cm

a Write down an algebraic expression for the height of the rectangle.

b Write down an algebraic expression for the area of the rectangle.

c Given that the area of the rectangle is 40 cm2, form and solve a quadratic equation to work out the value of t.

4 A rectangular garden is 4 m longer than it is wide. Its area is 165 m2.

a Sketch and label a diagram to show the area.

b Form and solve a quadratic equation to work out the dimensions of the garden.

5 I think of a negative number, square it and add five times the original number. My answer is 24. What number did I think of?

6 I think of a positive number.I square it, then subtract six times the number. The answer is 27. What was my original number?


ObjectivesA Solve quadratic equations by factorising

A* Factorise quadratic expressions of the form ax2 1 bx 1 c

Why learn this?By breaking down an algebraic

expression you can discover some of the properties of

the expression.

Skills check1 Write down all the pairs of numbers whose product is

a 10 b 12 c 2362 Factorise

a x2 1 3x 2 4 b x2 1 5x 1 6 c x2 2 8x 1 7

L

Factorising quadratics of the form ax2 1 bx 1 cIn the expression ax2 1 bx 1 c, the a is the coefficient of x2 and b is the coefficient of x.

In the quadratic expression 3x2 1 13x 1 4, the coefficient of x2 is 3. The first terms in the brackets must multiply to give 3x2.

The first terms in the brackets must be 3x and 1x. 3 is a prime number – the only factors are 3 and 1.

3x2 1 13x 1 4 5 (3x )(x )

The product of the last two terms must be 14.

Possible pairs of numbers are 1 and 4, 21 and 24, 2 and 2, or 22 and 22.

The coefficient of x is positive (113), so the two numbers must be positive.

Don’t forget negative numbers.

AO3

BAO2

B


Possible factorisations are(3x 1 4)(x 1 1) (3x 1 1)(x 1 4) (3x 1 2)(x 1 2)

Try expanding each one.(3x 1 4)(x 1 1) 5 3x2 1 3x 1 4x 1 4 7

(3x 1 1)(x 1 4) 5 3x2 1 12x 1 x 1 4 3 Always check using FOIL to expand the brackets.

(3x 1 2)(x 1 2) 5 3x2 1 6x 1 2x 1 4 7

So 3x2 1 13x 1 4 5 (3x 1 1)(x 1 4)

Factorising quadratics of the form ax2 1 bx 1 c when the coefficient of x2 is not primeWhen the coefficient of x2 is not prime, there are more possible cases to consider.

Exercise 20JFactorise each quadratic expression.

1 2x2 1 5x 1 3 2 3x2 1 14x 1 8 3 5x2 1 12x 1 4

4 7x2 1 26x 1 15 5 5x2 1 19x 2 4 6 3x2 2 4x 2 4

7 11x2 2 13x 1 2 8 2x2 2 5x 1 2 9 3x2 2 19x 1 20

10 5x2 2 39x 2 8 11 2x2 2 14x 1 24 12 7x2 2 8x 2 12

A

Factorise 2x2 2 7x 2 4.

Example 10

2x2 2 7x 2 4 5 (2x )(x )

Pairs of numbers whose product is 24 are 21 and 4, 1 and 24, or 2 and 22.

So the possible factorisations are

(2x 2 1)(x 1 4) 5 2x2 1 8x 2 x 2 4

(2x 1 4)(x 2 1) 5 2x2 2 2x 1 4x 2 4

(2x 1 1)(x 2 4) 5 2x2 2 8x 1 x 2 4

(2x 2 4)(x 1 1) 5 2x2 1 2x 2 4x 2 4

(2x 1 2)(x 2 2) 5 2x2 2 4x 1 2x 2 4

(2x 2 2)(x 1 2) 5 2x2 1 4x 2 2x 2 4

Therefore 2x2 2 7x 2 4 5 (2x 1 1)(x 2 4).

The only factors of 2 are 1 and 2.

One must be positive and one negative since the number term is negative.

This gives the 27x required. Always check that the x term is correct.

A

With more practice you will not need to write out all the combinations but will be able to work them out in your head.


Exercise 20K1 Factorise each quadratic expression.

a 8x2 1 17x 1 2 b 4x2 1 8x 1 3 c 6x2 1 17x 1 5

d 6x2 1 10x 1 4 e 8x2 1 20x 1 12 f 30x2 1 52x 1 16

2 Factorise each quadratic expression. You need to divide through by a common factor first.

a 2x2 1 8x 1 6 5 2(x2 1 1 ) b 3x2 1 21x 1 30

c 18x2 1 69x 1 60

A

Example 11Factorise 6x2 1 11x 1 4.

(3x )(2x ) or (6x )(x )

Pairs of numbers whose product is 14 are 2 and 2 or 1 and 4.

So the possible factorisations are

(3x 1 2)(2x 1 2) 5 … 1 6x 1 4x 1 … 5 … 1 10x 1 …

(6x 1 2)(x 1 2) 5 … 1 12x 1 2x 1 … 5 … 1 14x 1 …

(3x 1 1)(2x 1 4) 5 … 1 12x 1 2x 1 … 5 … 1 14x 1 …

(3x 1 4)(2x 1 1) 5 … 1 3x 1 8x 1 … 5 … 1 11x 1 …

(6x 1 1)(x 1 4) 5

(6x 1 4)(x 1 1) 5

Therefore 6x2 1 11x 1 4 5 (3x 1 4)(2x 1 1).

Factors of 6 are 1 and 6 or 2 and 3.

All terms are positive, so only consider positive numbers.

This gives 111x required.You can stop trying once you have found the correct pair.

A

Factorise 8x2 2 29x 2 12.

Example 12

8x2 2 29x 2 12 5 (8x )(x ) or (4x )(2x )

Pairs of numbers whose product is 212 are 212 and 1, 12 and 21, 26 and 2, 6 and 22, 24 and 3, 4 and 23.

Possible factorisations are(4x 2 12)(2x 1 1) 5 8x2 2 20x 2 12(8x 2 1)(x 1 12) 5 8x2 1 95x 2 12(8x 1 6)(x 2 2) 5 8x2 2 10x 2 12

The correct factorisation is (8x 1 3)(x 2 4) 5 8x2 2 29x 2 12.

There are many possible combinations. Try different ones until you find which one will give you 229x.

AO2

A


Exercise 20L1 Factorise each quadratic expression.

a 10x2 1 x 2 3 b 12x2 2 x 2 6 c 4x2 1 2x 2 6

d 20x2 1 19x 2 28 e 30x2 2 52x 1 16

2 Factorise each quadratic expression. You need to divide through by a common factor first.a 5x2 1 5x 2 10 b 14x2 1 35x 2 84

c 15x2 2 72x 2 15 d 28x2 2 88x 1 12

e 24x2 1 4x 2 4 f 50x2 2 70x 2 60 g 36x2 2 42x 1 12

A

A

AO3

A

Find the values of x which satisfy the equation 8x2 5 14x 1 4.

Example 13

Step 1: 8x2 2 14x 2 4 5 0

Step 2: (2x 2 4)(4x 1 1) 5 0

Step 3: 2x 2 4 5 0 or 4x 1 1 5 0 2x 5 4 or 4x 5 21

So x 5 2 or x 5 2 1 __ 4

Rearrange the equation to make one side zero.

Factorise the equation.

Solve the two linear equations.

A

Exercise 20M1 Find the roots of these quadratic equations.

Leave your answers as fractions where necessary.

a 2a2 1 5a 2 3 5 0 b 3b2 1 5b 1 2 5 0 c 4c2 2 c 2 5 5 0

d 0 5 5d2 2 8d 2 4 e 6e2 2 16e 1 8 5 0 f 4f 2 2 6f 2 4 5 0

g 6g2 1 19g 1 10 5 0 h 0 5 4h2 1 8h 1 4 i 7i2 2 3i 2 4 5 0

2 Find the values of x which satisfy these equations.

a 2x2 5 4x 1 6 b 9x2 1 10 5 21x c 10x2 1 13x 5 9

d 4x 1 16 2 6x2 5 0 e 15x2 5 230x 2 15 f (x 1 2)(x 2 2) 5 3x

3 a Write down an algebraic expression for the

3x

2x � 1

area of the rectangle.

b The area of the rectangle is 108 cm2. Form and solve an algebraic equation to find the value of x.

c What is the perimeter of the rectangle?

4 I think of a number.Three times the square of my number is equal to twelve times my number.Work out the possible values of my number.

AO2

A

30920.5 Using the quadratic formula

5 Next year Yvette will be four times her daughter Amelia’s age.Let x represent Amelia’s age next year.

a Write down an algebraic expression for i Yvette’s age next year ii Amelia’s age this year iii Yvette’s age this year.

b The product of their ages is 351. Form and solve an algebraic equation to work out Amelia’s age.

c How old is Yvette this year?


Skills check1 Using the formula x 5

3y ___

8 2 √

__ z , find the value of x when

a y 5 16, z 5 100 b y 5 24, z 5 49 c y 5 80, z 5 100

LWhy learn this?

This method solves quadratic equations

that you can’t factorise, like x2 1 3x 2 7.

ObjectivesA A* Solve quadratic equations by

using the quadratic formulaA* Decide how many solutions a quadratic equation has by

considering the discriminant

The quadratic formulaSometimes a quadratic expression cannot be factorised.

You can use the quadratic formula to solve a quadratic

You do not need to learn the formula – it will be on the exam formula sheet.

equation of the form ax2 1 bx 1 c 5 0, where a 0.

x 5 2b 6 √________

b2 2 4ac _______________ 2a

Be careful! You cannot use the quadratic formula until you have made one side of the equation zero.

Keywordsquadratic formula, discriminant

There will be two solutions.

Use the quadratic formula to solve the equation x2 1 3x 2 7 5 0.

Example 14

a 5 1, b 5 3, c 5 27

x 5 2b 6 √_________

b2 2 4ac _______________ 2a

5 23 6 √_________________

32 2 4 3 1 3 27 _________________________ 2 3 1

5 23 6 √________

9 1 28 _______________ 2

5 23 6 √___

37 ___________ 2

x 5 23 1 √___

37 ___________ 2 or x 5 23 2 √___

37 ___________ 2

Write down the values of a, b and c.

Substitute these values into the quadratic formula. Be careful with the negative value.

Leave your answer in surd form.

Simplify the calculation. Follow the order of operations.

A

AO2

A*


A

AO2

A

A

Exercise 20NUse the quadratic formula to solve each equation. Leave your answers in surd form.

1 x2 1 3x 2 9 5 0 2 x2 1 5x 2 12 5 0

3 x2 1 6x 1 5 5 0 4 x2 1 6x 1 2 5 0

5 3x2 2 2x 2 8 5 0 6 2x2 1 8x 2 20 5 0

7 5y2 1 12y 2 4 5 0 8 12r2 2 8r 1 1 5 0

9 7t2 2 2t 2 8 5 0 10 3g2 1 7g 1 3g 5 0

Solve the quadratic equation 2x2 5 6x 1 12.

Example 15

2x2 2 6x 2 12 5 0

a 5 2, b 5 26, c 5 212

x 5 2b 6 √_________

b2 2 4ac _______________ 2a

5 2 (26) 6 √

_______________________

(26)2 2 4 3 2 3 (212) _________________________________ 2 3 2

5 6 6 √_________

36 1 96 _______________ 4

5 6 6 √_____

132 __________ 4

5 6 6 2 √___

33 __________ 4

5 3 6 √___

33 _________ 2

First rearrange the equation to make one side zero.

Be very careful with positive and negative numbers.

A

Divide all terms by 2.

√____

132 5 √______

4 3 33 5 2 √___

33

Exercise 20O1 Use the quadratic formula to solve these equations.

Make sure you rearrange the equations first.

Leave your answers in surd form.

a x2 5 4x 1 1 b x2 1 16 5 12x c x2 2 8x 5 6

d x2 5 1 2 6x e 4 1 2x 5 x2 f x2 1 8x 1 2 5 0

2 Try to solve this quadratic equation using the quadratic formula.x2 1 x 1 1 5 0

Explain why you cannot find a solution.

3 A carpet manufacturer wishes to make carpet tiles with area 1500 cm2.The tiles are rectangular and the length is 10 cm less than double the width.Work out the dimensions of a carpet tile.

4 Look at your answer to Q3. Did you need to use the quadratic formula?Explain your answer.AO3

A*


The discriminantQuestion 2 in Exercise 20O asked you to try to solve the equation x2 1 x 1 1 5 0.

Using the quadratic formula

a 5 1, b 5 1, c 5 1

x 5 2b 6 √________

b2 2 4ac _______________ 2a

5 21 6 √______________

12 2 4 3 1 3 1 _____________________ 2 3 1

5 21 6 √___

23 __________ 2

The calculations result in trying to find the square root of a negative number. This has no real solutions – you will learn more about this if you do A-level maths.

b2 2 4ac in the quadratic formula is known as the discriminant.In general,

• when b2 2 4ac . 0, there are two distinct solutions to the quadratic equation

• when b2 2 4ac , 0, there are no real solutions to the quadratic equation

• when b2 2 4ac 5 0, there is one solution (sometimes called a repeated root).

By considering the discriminant, decide whether each of these quadratic equations has zero, one or two solutions.

a 3x2 1 2x 2 5 5 0

b 7x2 5 10x 2 8

c 9x2 1 16 5 24x

Example 16

a 3x2 1 2x 2 5 5 0

a 5 3, b 5 2, c 5 25

b2 2 4ac 5 22 2 4 3 3 3 25

5 4 1 60

5 64

Since 64 . 0 there are two solutions.

b 7x2 5 10x 2 8

7x2 2 10x 1 8 5 0

a 5 7, b 5 210, c 5 8

b2 2 4ac 5 (–10)2 2 4 3 7 3 8

5 100 2 224

5 2124

Since 2124 , 0 there are no solutions.

Write down the values of a, b and c.

Work out b2 2 4ac

Rearrange to the form ax2 1 bx 1 c 5 0

A*


c 9x2 1 16 5 24x

9x2 2 24x 1 16 5 0

a 5 9, b 5 224, c 5 16

b2 2 4ac 5 (224)2 2 4 3 9 3 16

5 576 2 576

5 0

There is one (repeated) solution.

Rearrange to the form ax2 1 bx 1 c 5 0

Exercise 20P1 For each quadratic equation, decide if there are zero, one or two solutions.

a 3x2 1 2x 2 4 5 0 b 5m2 1 9m 1 6 5 0 c 3t2 1 6t 1 3 5 0

d 4d2 2 5d 1 6 5 0 e 0 5 2z2 1 5z 1 1 f 4x2 5 3x 2 1

g 9t 5 5t2 2 12 h 2q2 5 2 8q 2 8

A*

L20.6 Completing the square

Why learn this?In mathematics, as in

life, it is important to have more than one way

to solve a problem.

Skills check1 Expand and simplify

a (x 1 2)2 b (x 2 3)2 c (x 2 5)2

ObjectivesA* Solve a quadratic equation by completing

the square

Completing the squareCompleting the square is another way to solve a quadratic equation which cannot be factorised.

Expanding an expression of the form (x 1 a)2 gives

(x 1 a)(x 1 a) 5 x2 1 2ax 1 a2

Working backwards, this can be used to ‘complete the square’.

Consider the equation x2 1 4x 1 10 5 0.

For the coefficient of x to be 4 the squared bracket must be (x 1 2)2.

But (x 1 2)2 5 x2 1 4x 1 4.

To get from (x 1 2)2 to x2 1 4x 1 10 you need to subtract 4 and then add 10.x2 1 4x 1 10 5 0(x 1 2)2 2 4 1 10 5 0(x 1 2)2 1 6 5 0

This is half the coefficient of x.

HELP Section 12.5

31320.6 Completing the square

Exercise 20Q1 Write each expression in completed square form.

a x2 1 6x 1 3 b x2 1 2x 1 7 c x2 2 8x 1 5

d x2 2 12x 1 12 e x2 2 4x 2 7 f x2 2 10x 2 1

2 Write each algebraic expressions in the form Be careful with the values of p and q. Are they positive or negative?

(x 1 p)2 1 q, giving the values of p and q.

a x2 1 10x 1 32 b x2 1 2x 1 2 c x2 2 4x 1 20

d x2 2 14x 1 10 e x2 2 6x 2 3 f x2 2 4x 2 2

A*

Write the expression x2 1 10x 1 9 in completed square form.

Example 17

(x 1 5)2 5 x2 1 10x 1 25

The expression required is x2 1 10x 1 9

5 (x2 1 10x 1 25) 2 25 1 9

5 (x 1 5)2 2 25 1 9

5 (x 1 5)2 2 16

Halve the coefficient of x.

Subtract the square of the number in the bracket.

Put in the original number term.

A*

By completing the square, solve the equation x2 2 8x 1 5 5 0.

Leave your answer in surd form.

Example 18

x2 2 8x 1 5 5 (x 2 4)2 2 16 1 5

5 (x 2 4)2 2 11

So (x 2 4)2 2 11 5 0

(x 2 4)2 5 11

x 2 4 5 6 √___

11

x 5 4 6 √___

11

Write the expression in completed square form.

Solve the equation by rearranging.

A*

This is the exact answer in surd form.

Exercise 20R1 Solve the quadratic equations by completing the square.

a x2 1 10x 1 9 5 0 b x2 1 2x 2 8 5 0

c x2 2 8x 1 10 5 0 d x2 2 12x 1 16 5 0

e x2 2 4x 2 4 5 0 f x2 1 6x 2 7 5 0

A*


2 Give the exact solution to these quadratic equations by completing the square. Leave your answers in surd

form where apprpriate.a x2 1 8x 2 9 5 0 b x2 1 4x 2 8 5 0

c x2 2 2x 2 1 5 0 d x2 2 8x 1 10 5 0

e x2 2 20x 1 50 5 0 f x2 2 14x 1 41 5 0

3 Solve these quadratic equations Don’t forget to rearrange the equations first. by completing the square.

a x2 5 6x 2 4 b 3x(x 1 6)5 6

c (x 1 1)(x 2 5) 5 7 d (x 2 2)(x 1 8) 5 7

e 2 _____ x 1 6

5 x f 3 ____________ (r 2 1)(r 1 2)

5 1

Review exercise1 Factorise the expression x2 1 6x 1 5. [2 marks]

2 I think of a number, square it, then subtract three times the number. The result is 108.Form and solve an algebraic equation to work out the possible values of the number I thought of. [4 marks]

3 I think of a number, square it, then add it to 5 times the number.The answer is 24. Form and solve an algebraic equation to work out the possible values of the number I thought of. [4 marks]

4 a Factorise 2x2 2 15x 2 8. [2 marks]

b Hence solve the equation 2x2 2 15x 2 8 5 0. [2 marks]

5 Factorise 6y2 1 13y 2 5. [2 marks]

6 Use the quadratic formula to solve 2x2 2 6x 1 1 5 0.Leave your answer in surd form. [3 marks]

7 a Factorise the quadratic expression 6x2 2 11x 2 10. [2 marks]

b Hence solve the equation 6x2 2 11x 2 10 5 0. Leave your answers as fractions. [2 marks]

8 A rectangular piece of land has length 3 m more than double the width. The area of the rectangle is 170 m2. Work out the dimensions of the rectangle. [5 marks]

9 A rectangular rug is 6 m longer than its width. The area of the rug is 16 m2. Calculate the dimensions of the rug. [5 marks]

A*

AO3

B

B

A

AO3

A*

315Chapter 20 Summary

10 a Find the values of m and n such that x2 1 4x 2 6 5 (x 1 m)2 2 n. [2 marks]

b Hence solve the equation x2 1 4x 2 6 5 0 by rearranging, leaving your answer in the form a 6 √

__ b . [3 marks]

11 How many roots does each of these quadratic equations have?

a 5x2 2 2x 2 7 5 0 b 3x2 2 11x 1 12 5 0 c 4x2 2 12x 1 9 5 0 [6 marks]

12 a Write the following algebraic expression in completed square form.x2 2 4x 1 2 [2 marks]

b Hence find the exact solution to the equation x2 2 4x 1 2 5 0. [2 marks]

A*

In this chapter you have learned how to

• factorise a quadratic expression of the form x2 1 bx 1 c B

• solve quadratic equations by rearranging B

• factorise a quadratic expression that is the difference of two squares B A

• solve quadratic equations by factorising B A

• factorise quadratic expressions of the form ax2 1 bx 1 c A A*

• solve quadratic equations by using the quadratic formula A A*

• decide how many solutions a quadratic equation has by considering the discriminant A*

• solve a quadratic equation by completing the square A*

Chapter summary

quadratic equations · 2017-07-12 · 1 factorise a 3x 1 6y b 8x 1 x2 c 3m2 1 mn ... b what do you...

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