Quadratic equations Lesson 2 Source : Lial, Hungerford and Holcomb (2007), Mathematics with...

Quadratic equationsLesson 2

Source :Lial, Hungerford and Holcomb (2007), Mathematics with Applications, 9th edition, Pearson Prentice Hall, ISBN 0-321-44947-9 (Chapter 1 pp.56-64)

Quadratic Equations

Quadratic equations could be of two types:

1. Complete

Where a, b, and c are real numbers and a 0

2. Incomplete:

02 bxax 02 cax02 ax

02 cbxax

Quadratic Equations: Incomplete Quadratic Equations

Solving Incomplete Quadratic Equations:

02 bxax

3;0:

0)3(;02

0)3(2

062

:2

xxAnswer

xx

xx

xx

Example

Quadratic Equations: Incomplete Quadratic Equations

Solving Incomplete Quadratic Equations:

02 cax

2;2:

4

82

082

:

2

2

2

xxAnswer

x

x

x

Example

Quadratic Equations: Complete Quadratic Equations

Solving Complete Quadratic Equations:

02 cbxax

Three methods to solve:

1. Using Discriminant

2. Viete’s Theorem

3. Completing the square

Quadratic Equations: Complete Quadratic Equations

Solving Complete Quadratic Equations: 1.Using Discriminant:

02 cbxax

antdiscriacbD

a

acbbx

Solution

min4

2

4

:

2

2

2,1

3. Discriminant can be negative and we'd get no real solutions.

The "discriminat" tells us what type of solutions we'll have.1. Discriminant can be positive and we'd get two unequal real

solutions 04D 2 acb

2. Discriminant can be zero and we'd get one solution (called a repeated or double root because it would gives us two equal real solutions).

04D 2 acb

04 D 2 acb

Quadratic Equations: Complete Quadratic Equations

Example 3. Discriminant can be negative and we'd get no real solutions.

Feel the power of the formula!

Example 1. Discriminant can be positive and we'd get two unequal real solutions 08152 2 xx

Example 2. Discriminant can be zero and we'd get one solution (called a repeated or double root because it would gives us two equal real solutions).

0962 xx

053x2 x

Quadratic Equations: Complete Quadratic Equations

Quadratic Equations: Complete Quadratic Equations

028122 xx We will complete the square

First get the constant term on the other side28122 xx

___ 28___ 122 xx

We are now going to add a number to the left side so it will factor into a perfect square.

36 36 6436 122 xx

Solving Complete Quadratic Equations: 2. Completing Square

6436 122 xx 646 2 xThis can be written as

Now we'll get rid of the square by square rooting both sides.

646 2 x Remember you need both the positive and negative root!

86 x Add 6 to both sides to get x alone.

86 x

14861 x 2862 x

Quadratic Equations: Complete Quadratic Equations

011244 2 xx

Example: Solve by completing the square:

Quadratic Equations: Complete Quadratic Equations

2

1;

2

11:

5)62(5)62(

25)62(

25)62(

6116622)2(

11622)2(

11244

011244

21

2

222

2

2

2

xxAnswer

xandx

x

x

xx

xx

xx

xx

Solution:

Quadratic Equations: Complete Quadratic Equations

According to Viete’s theorem if x1 and x2 are the solutions of

then

Solving Complete Quadratic Equations: 3.Viete’s theorem

02 cbxax

Quadratic Equations: Complete Quadratic Equations

074392 xx

Example 1: Solve by Viete’s theorem:

Example 2: Solve by Viete’s theorem:

0210573 2 xx

Quadratic Equations: Complete Quadratic Equations

)37,2(:

39

74

07439

21

21

2

Answer

xx

xx

xx

Example 1: Solve by Viete’s theorem:

Quadratic Equations: Complete Quadratic Equations

Example 2: Solve by Viete’s theorem:

)5,14(:

193

57

703

210

0210573

21

21

2

Answer

xx

xx

xx

Solving by reducing to quadratic equations:

Example 1: Reduce to quadratic equation and solve

Practice yourself: Reduce to quadratic equation and solve

023 24 xx

Some higher degree equations could be reduced to quadratic equations and then solved

087 ) 36 xxa

084)5(8)5( ) 222 xxxxb

Solving by reducing to quadratic equations:Example 1: Solution

)2,2,1,1(x :Answer

2 2 2

1 1 1

,

3

2

023

:expression theRe

: variablenew Introduce

023

4321

3,22

22

2,12

11

21

21

2

2

24

xxx

xxyy

xxyy

Hence

yy

yy

yy

write

xy

xx

Important!If we know the roots of quadratic equation

we could factorize any quadratic expression easy way:

Example 2: Solve by Viete’s theorem:

Factorizing using roots

Example 1: Factorizing using roots

Solution:

028122 xx

141 x 22 x

Factorizing:

)2)(14())((2812 212 xxxxxxaxx

Factorizing using roots

Example 2: Factorizing using roots

Solution:

51 x 142 x

Factorizing:

)14)(153()14)(5(3

)14)(5(3))((210573 212

xxxx

xxxxxxaxx

0210573 2 xx