2. Equations & Inequalities

Factorizing Quadratic Equations

The Quadratic Formula

“k”- Substitution

Simultaneous Equations

Completing the Square

Word Problems

Quadratic Inequalities

The Number System

Nature of Roots 1

Factorizing Quadratic Equations

Standard form: ax² + bx + c = 0

Solve for x:

1) x² = 25

=

x = + 5

or x² - 25 = 0

(x – 5)(x +5) = 0

x = 5 or x = -5

Difference of Squares

Two answers when you take the square root 2

Solve for x:

2) x² - 5x + 4 = 0 Trinomial

(x-4)(x-1) = 0

x = 4 or x = -1

3) 2x(x - 2) = 16

2x² - 4x = 16

2x² - 4x – 16 = 0

2(x² - 2x – 8) = 0

2(x – 4)(x+2) = 0

x = 4 or x = -2

First take out a

common factor

before factorizing

the trinomial

3

Solve for x:

4) x² - 2x = 1

x² - 2x – 1 = 0

This trinomial can’t be

factorized …

Now what ???

Standard form of a

Quadratic Equation

4

THE Quadratic Formula

The quadratic equation must always be in

standard form i.e. 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0

To solve for 𝑥, substitute 𝑎, 𝑏 𝑎𝑛𝑑 𝑐 into THE

Quadratic Formula …

𝑥 =−𝑏 ± 𝑏2 − 4𝑎𝑐

2𝑎

5

4) x² - 2x = 1

x² - 2x – 1 = 0

So, if we look at the previous example …

ax² => a = 1

bx => b = -2

c => -1

Quadratic Formula Calculator

𝒙 =−𝒃 ± 𝒃𝟐 − 𝟒𝒂𝒄

𝟐𝒂

6

Solve for x:

5)

LCD: x² + 2x =>

complicated!

Let x² + 2x = k

k² - 6 = k

k² - k – 6 = 0

(k - 3)(k + 2) = 0

k = 3 or k = -2

Times through by

LCD = k

Have we solved

for x yet?

Rather let repeated

expression = k

“k”- Substitution

7

So, we said … Let x² + 2x = k

AND we’ve just solved for k … k = 3 or k = -2

x² + 2x = k

x² + 2x = 3

x² + 2x - 3 = 0

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

x = -3 or x = 1

OR …

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x² + 2x = k

x² + 2x = -2

x² + 2x + 2 = 0

Can’t be

factorized …

The Quadratic

Formula!

a = 1

b = 2

c = 2

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Simultaneous Equations

1. Solve for x and y: 2y = 4x

y² - 3y – x = 2

2y = 4x

y = 2x

1

Subst. into

y² - 3y – x = 2

(2x)² - 3(2x) – x = 2

4x² - 6x – x – 2 = 0

4x² - 7x - 2 = 0

(4x + 1)(x – 2) = 0

2

1

3

3 2

x = - ¼ or x = 2

Are we done

yet? 10

Solve for x and y: 2y = 4x

y² - 3y – x = 2

1

Subst. x = - ¼ or x = 2 into simplest equation

2y = 4x

2y = 4(- ¼ )

2y = -1

y = - ½

2

or 2y = 4x

2y = 4(2)

2y = 8

y = 4

Don’t forget to solve for y!

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Simultaneous Equations

2. Solve for x and y: -2 + y – 2x = 0

xy = 12

-2 + y – 2x = 0

y = 2x + 2

1

Subst. into

xy = 12

x(2x+2) = 12

2x² + 2x – 12 = 0

x² + x - 6 = 0

(x + 3)(x – 2) = 0

2 1

3

3 2

x = -3 or x = 2

Are we done

yet? 12

Solve for x and y: -2 + y – 2x = 0

xy = 12

1

Subst. x = -3 or x = 2 into simplest equation

xy = 12

(-3)y = 12

y = -4

2

or xy = 12

(2)y = 12

y = 6

Don’t forget to solve for y!

Solving a hyperbola

and a straight line

Solving a circle

and a straight line 13

Solve for x by Completing the Square

1. x2 - 8 x – 7 = 0

x2 – 8x = 7

x2 - 8x + ( −8

2 )2 = 7 + (

−8

2 )2

x2 - 8x + (-4)² = 7 + 16

(x – 4) 2 = 23

x – 4 = ± 23

x = 4 ± 23

x = 8,80 or x = -0,80

Graphical Representation of Completing the Square

Add half the

coefficient of

x & square it

3 terms form the

perfect square

Take the

square root

Two answers!

Completing the Square Method 14

2. 3x2 - 12x - 9 = 0

x2 – 4x - 3 = 0

x2 – 4x = 3

x2 - 4x + ( −4

2 )2 = 3 + (

−4

2 )2

x2 - 4x + (-2)² = 7 + 4

(x – 2) 2 = 11

x – 2 = ± 11

x = 2 ± 11

x = 5,32 or x = -1,32

Add half the

coefficient of

x & square it

3 terms form the

perfect square

Take the

square root

Two answers!

NB! Coefficient of x2 must

be 1 before completing

the square.

Practice Completing the Square 15

Exercise

Solve for x by completing the square:

1. x2 + 6x + 1 = 0

2. x2 – 5x = - 3

3. 2x2 + 8x – 4 = 0

4. 3x2 = 9x - 2

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Completing the Square & The

Quadratic Formula

The Quadratic Formula Proof

Solving Quadratic Equations

Different Methods of Solving Quadratic Equations

Quadratic Equation Problems

17

Quadratic Word Problems

1. A farmer has a chicken enclosure which

has an area of 12m2. If he increases the

length by 5m and the breadth by 1m, then the

area of the enclosure is 3 times larger than

the original enclosure.

a) If the length is x metres,

write down an expression

for the breadth, in terms of x.

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a) If the length is x metres, write down an

expression for the breadth, in terms of x.

𝐴 = 𝑙 × 𝑏

12 = 𝑥 × 𝑏

𝑏 = 12

𝑥

b) Write down an equation which models this

scenario.

𝑙 + 5 𝑏 + 1 = 12 × 3

𝑥 + 512

𝑥+ 1 = 36

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c) Determine the length of the enclosure, if it

is given that x < 10.

𝑥 + 512

𝑥+ 1 = 36

12 + 𝑥 + 60

𝑥+ 5 = 36

𝑥 + 60

𝑥− 19 = 0

𝑥2 − 19𝑥 + 60 = 0

𝑥 − 4 𝑥 − 15 = 0

𝑥 = 4 𝑜𝑟 𝑥 = 15

But! 𝑥 ≠ 15 𝑎𝑠 𝑥 < 10, 𝑠𝑜 𝑥 = 4 ∴ 𝑙𝑒𝑛𝑔𝑡ℎ = 4𝑚.

20

2. A farmer stays 120 km from the biggest

city. If he travels by car, he travels 20 km/h

faster than by train and he saves 18 minutes.

Determine how fast the farmer travels by car.

Time (train) – T (car) = 18 minutes = 3

10 ℎ𝑜𝑢𝑟

120

𝑥 −

120

𝑥 + 20=

3

10

1200 𝑥 + 20 − 1200𝑥 = 3𝑥 𝑥 + 20

1200𝑥 + 24000 − 1200𝑥 = 3𝑥2 + 60𝑥

𝑥2 + 20𝑥 − 8000 = 0

𝑥 + 100 𝑥 − 80 = 0

𝑥 = −100 𝑜𝑟 𝑥 = 80

𝑇𝑖𝑚𝑒 =𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑆𝑝𝑒𝑒𝑑

∴ 𝐹𝑎𝑟𝑚𝑒𝑟 𝑡𝑟𝑎𝑣𝑒𝑙𝑠 𝑎𝑡 80𝑘𝑚/ℎ

Speed,

Distance &

Time Word

Problem

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