chapter 9: quadratic equations · 2017-02-12 · we previously have used factoring to solve...

Concept Category 4

Quadratic

Equations

§ 1

Solving Quadratic

Equations by the Square

Root Property

Martin-Gay, Developmental Mathematics 3

Square Root Property

We previously have used factoring to solve

quadratic equations.

This chapter will introduce additional

methods for solving quadratic equations.


If b is a real number and a2 = b, then

ba


Solve x2 = 49

2x

Solve (y – 3)2 = 4

Solve 2x2 = 4

x2 = 2

749 x

y = 3 2

y = 1 or 5

243 y


Example


Solve x2 + 4 = 0

x2 = 4

There is no real solution because the square root

of 4 is not a real number.


Example


Solve (x + 2)2 = 25

x = 2 ± 5

x = 2 + 5 or x = 2 – 5

x = 3 or x = 7

5252 x


Example


Solve (3x – 17)2 = 28

72173 x

3

7217 x

7228 3x – 17 =


Example

§ 2

Solving Quadratic

Equations by the

Quadratic Formula


The Quadratic Formula

Another technique for solving quadratic

equations is to use the quadratic formula.

The formula is derived from completing the

square of a general quadratic equation.


A quadratic equation written in standard

form, ax2 + bx + c = 0, has the solutions.

a

acbbx

2

42



Solve 11n2 – 9n = 1 by the quadratic formula.

11n2 – 9n – 1 = 0 set one side = 0

a = 11, b = -9, c = -1

)11(2

)1)(11(4)9(9 2

n

22

44819

22

1259


Example


Two kinds of answers:

Decimal Answers (to graph):

Simplified Radical Answers (SAT, ACT,

and other college placement exams):

9 125 9 11.2 9 11.2 9 11.2

22 22 22 22

0.9 0.1

and

and

9 5 25 9 5 5

22 22


Practice: Solve x using QFormula

Present your answers in

Decimals and Simplified Radicals:

2

2

] ( ) 4 12 63

b] 2 12 46

a f x x x

y x x


1/18 Practice Now:

Vertex point?

x-intercept points?

y-intercept point?

2 2] ( ) 10 21 ] ( ) 2( 3) 5a f x x x b g x x

*Vertex point?

*x-intercept points?

*y-intercept point?


)1(2

)20)(1(4)8(8 2

x

2

80648

2

1448

2

128 20 4 or , 10 or 2

2 2

x2 + 8x – 20 = 0 multiply both sides by 8

a = 1, b = 8, c = 20

8

1

2

5Solve x2 + x – = 0 by the quadratic formula.

Quadratic Formula – SAT example

Example


Concept Category 4 Quadratics

Standard Form of Quadratic Equation

Vertex Form of Quadratic Equation

Vertex: transformation first

Radical Operations

Solving Radical Equations

Nth Roots

Complex Numbers

int : Factoring or Quad: ratic Formula2

bVertex x

a


4

16

25

100

144

= 2

= 4

= 5

= 10

= 12

Radicals


Perfect Squares

1

4

9

16

25

36

49

64

81

100

121

144

169

196

225

256

324

400

625

289


8

20

32

75

40

=

=

=

=

=

2*4

5*4

2*16

3*25

10*4

=

=

=

=

=

22

52

24

35

102

Perfect Square Factor * Other Factor

LE

AV

E I

N R

AD

ICA

L F

OR

M


Simplify each expression

737576 78

62747365 7763

A]

B]


+ To combine radicals: combine the

coefficients of like radicals


Simplify each radical first, then combine.

323502 2 25* 2 3 16 * 2

2 *5 2 3* 4 2

10 2 12 2

2 2


Practice NOW

485273 3 9 * 3 5 16 * 3

3* 3 3 5* 4 3

9 3 20 3

29 3


* Multiply the coefficients and then

multiply the radicands and then simplify


35*5 175 7*25 75

Multiply and then simplify

73*82 566 14*46

142*6 1412

204*52 8 100 8*10 80


2X

6Y

264 YXP

244 YX

10825 DC

= X

= Y3

= P2X3Y

= 2X2Y

= 5C4D5


3X

XX

=

=

XX *2

YY 45Y

=

= YY 2


33YPX

2712 YX

9825 DC

=

=

= 5Y

PXYYX *22

5Y

PXYXY=


Happy Wed. 1/25

Did everyone have a chance to work on

yesterday’s SMC practice test?

Placement Exams for English and Math are

required for ALL CA colleges (2 or 4 yr)

This practice test have other parts: 2nd part of

Algebra, Geometry, and Precalculus

The less you score and more courses ($$$$ +

Time) you will need to make up for


CA College Placement Exams v.s. CAASP

Why they all have similar questions?

Because your HIGH SCHOOL

standards/State Exams are mostly

set by public colleges


2

5 5*5 25 5

4

3 7 3 7 *3 7 *3 7 *3 7 81*49

3969

2

48w 4 48 * 8w w 864w

48w

2

2 3x 2 3 *2 3x x 24 9x 12x


2 2

3 5 4 20 3 45 4 3

Practice NOW

4

43 2 2 16x x


SOLUTIONS

2

3 5

316x


A] Divide the

coefficients, divide the

radicands if possible,

B] rationalize the

denominator so that the

denominator is always

an integer!


Fractional Radicands

22

3

2

2

2 4

3 9 Simplify exponents/radicals

first. Then reduce.

4

9

4 2

39

20

36 20

36

4 5 2 5 5

6 6 3


7

568 2*4 22


15 15

749

You Try:

63

812

45

16

15

49

45 9 5 3 5

416 16

2 63 2 9 7 2 3 7 6 7 2 7

9 9 9 381


How about….

50

3

5 2

3

When simplifying radicals, there can never be a radical in the denominator of the final answer.

BIG

NO NO!

Rationalization: the steps to change a radical denominator to a whole number.


24 4 * 6 2 6

28 4 * 7 2 7

Method:

Multiply the

Denominator

by the same

radical

7

6

7*

7

42

49

7

42

In the world of SAT, ACT, math placement exams:

Fraction

expansion


This can be

divided which

leaves the

radical in the

denominator.

We do not leave

radicals in the

denominator. So

we need to

rationalize.

10

5 1

2*

2

2

2

2


12

3 3

2 3*

3

3

3 3

2 9

3 3

6

3

2

Reduce the fraction.

Rationalize


Try it yourself. Simplify the following.

7 14

49

6

3y

x

98

7

7 2

7

77 2

7 7

7 14

7 14

How about with variables…

1)

2) x3

y y

3 yx

y y y

3

2

x y

y y

3x y

y y

3

2

x y

y


12

3 4 * 3

3 4 2

4

5

4x

x

2

2

2x

x x

2 x

x x

2 x

x

3

13

3 18

4

x x

x

6

3 *3 2

2

x x x

x x

4

9 2

2x

2 5 3

6

3 28

12

xy x y

x

2 4 2

6

3 4 *7???

4 *3

xy x xy y

x


Happy MONDAY !

Quick Check on Wedn. This week:

Will be on gradebook

*Standard Form: vertex point, x and y intercepts, sketch

*Vertex Form: vertex point, x and y intercepts, sketch

*Radicals: Simplify, add, subtr, multi, div (rationalization)

*Solving Radical equations


Quick Check Tomorrow


Self-Evaluation, Correction, Practice

1) Self Evaluation: Mark right or wrong with….

“CO” Conceptual Errors

i.e. wrong formulas, wrong solving steps

“CA” computational Errors

2) Write the actual correction work on separate paper

Turn in (1) and (2) today

3) Practice: use the other version


Simplify

5 6 5 3

5 3 5 3

5 6

5 3

When radicals have binomial radicals in the denominator, multiply the numerator and denominator by the conjugate of the denominator to eliminate the radical in the denominator.

Simplify by distributing or using FOIL. then continue to completely simplify.

5 3 5 6 5 18

5 9

23

4

59

Be sure to simplify all numbers outside the radical if they all have common factors.


Rationalizing Binomial Radicals

6

3 2

3 6 2 3

7

3 5

4 3

12 3 3 4 5 15

13


Solve x(x + 6) = 30 by the quadratic formula.

x2 + 6x + 30 = 0

a = 1, b = 6, c = 30

)1(2

)30)(1(4)6(6 2

x

2

120366

2

846

This is Complex Number !!!!


Example


The expression under the radical sign in the

formula (b2 – 4ac) is called the discriminant.

The discriminant will take on a value that is

positive, 0, or negative.

The value of the discriminant indicates two

distinct real solutions, one real solution, or no

real solutions, respectively.

The Discriminant


Use the discriminant to determine the number and

type of solutions for the following equation.

5 – 4x + 12x2 = 0

a = 12, b = –4, and c = 5

b2 – 4ac = (–4)2 – 4(12)(5)

= 16 – 240

= –224

There are no real solutions.

The Discriminant

Example


Solving Quadratic Equations

Steps in Solving Quadratic Equations

1) If the equation is in the form (ax+b)2 = c, use

the square root property to solve.

2) If not solved in step 1, write the equation in

standard form.

3) Try to solve by factoring.

4) If you haven’t solved it yet, use the quadratic

formula.


Solve 12x = 4x2 + 4.

0 = 4x2 – 12x + 4

0 = 4(x2 – 3x + 1)

Let a = 1, b = -3, c = 1

)1(2

)1)(1(4)3(3 2

x

2

493

2

53

Solving Equations

Example


Solve the following quadratic equation.

02

1

8

5 2 mm

0485 2 mm

0)2)(25( mm

02025 mm or

25

2 mm or

Solving Equations

Example


2/7/17 Complex Number


What imaginary

roots actually

look like on

a graph (which

is why we don’t

usually graph them)


i

1 i

Before, is no solution.

But now a special symbol is assigned

so we can carry on the computation.

negative


Examples 1i

16 16 1

4 i 4i

162 2 81 1

2 9 i

9 2i


2i 1 1 i i 1 !ALWAYS

3i i i i 2i i 1i i

4i 2 2 1 1iiii i i 1

5

6

i

i

2 2

2 2 2 1 ...

iiiii i i i i

iiiiii i i i the pattern repeats

1 !i Always


26 2 i i

Simplify

Divide the exponent by 4 (or 2) and look at the

remainder.

1

26 4=6 with remainder 2

Method

1

i26

13

2 13

2

26 2=13 ( 1) 1

Method

i


a] 8 3i i 24i2

24 1

Simplify each expression.

24

b] 5 20 i 5 i 20Remember that 1 i

i2 100 110 10


When adding or subtracting complex numbers,

combine like terms.

Ex: 8 3i 2 5i

8 2 3 5i i

10 2i


Multiplication:

8 5i 2 3i

16 24i 10i 15i2

F O I L

1614i 15

3114i


Complex Numbers Warm-Ups 8 minutes

30 45

2

:

}

b} (3 4 )

c} (3 4 )(3 4 )

d} (4 )(4 )

Multiply and Simplify

a i i

i

i i

i i


5]

3Ex

12]

36ex

i

5

3

3

3

5 3

3

1

3i

i

i

23

i

i

3

i

3

i

18 3]

27

iTry

i

6

9

i

i

6

9

i i

i i

2

2

6 1 6

9 9

i i i

i

Division & Rationalizing the Denominator

Denominator can only

be an integer !


Warm-up 8 minutes

202 55

:

]

] (4 5 )(4 5 )

] (2 6 )(2 6 )

Simplify

a i i

b i i

c

Solve using Quadratic Formula, what is the discriminant?

2] 2 2d y x x


Radical expression and Exponents

Exponents v.s. Radicals (Roots)

35 125

3 125 5therefore


Other examples

4

4 4

16

1

2 2 2 2

2 2

2

26 2 2

3

3 3

8

8

2 2 2

2 2 2

2

2

2

2 2

4

4

2

2 2

2 2

24


2] 64a

3

4

5

6

7

b] 64

] 64

] 64

] 64

] 64

c

d

e

f

2 2 22 2 2 2 32 8

3

4

5

6

7

2 2 2

2 2 2 2

2 2 2 2 2

2 2

2 2

2

2 2 2 2 2

2 2 2 2

2 2

2

2

2

4

5

7

2 4

4

2

2

64

2

2

More Examples:

only 6 of them, not enough…


2] 81a

3

4

5

b] 81

] 81

] 81

c

d

2 3 3 3 3 23 9

3

4

5

3

3 3 3

3 3 3

3 3 3

3

3

Not enough numbers

3

3

3 3

3

81still

More Examples:


1

33

3

3

8 8 8 8 512

8 8 2 2 2 2

1

44 4

416 16 16 16 16 65536

16 16 2 2 2 2 2

2

6729Try

on calculator

9 ?did you get

Rational Exponents

16 ^ ( 14 )

Rational Exponent is

actually for calculators

yx


Practice NOW!

103 240

2

] (2 4 )(2 7 )

] (2 3 )(2 3 )

] (2 3 )(2 3 )

]

] (2 8

1

)

:DOK Simplif

a i i

b i i

c

i

i

y

d i

e

2

2

2

2 :

] ( ) 2 8 3

] ( ) 6 10

? int?

y int? sk

]

e

] 2 3

2( 4) 6 8

tch

c

DOK

a f x x x

b g x x x

vert

d Solv

Solv

e

e x

x

x

e x

x


Answers

] 24 22

]13

]1

] 1

] 60 32

a i

b

c

d i

e i

( 2, 5)

8 40int

4

0.42, 3.58

int (0,3)

vertex

x

y

(3,1)

6 4int

2

6 23

2

int (0,10)

vertex

x

ii

y


2 53 2

ii

When you have a Binomial for Denominator:

3 23 2

ii

2

2

6 4 15 10

9 6 6 4

i i i

i i i

16 11

13i

6 7

6 7

24 4 7

36 6 7 6 7 7

24 4 7

29

4

6 7

16 1113 13

i

a.

b.

Rationalization for Binomials


Ex: Solve x2+ 6x +10 = 0

2 4

2

b b acx

a

a =

b =

c =

26 6 4 1 10

2 1

1st

6 36 4 1 10

2 1

2nd 6 36 40

2

6 4

2

6 2

2

i

6 2 6 2

2 2

i iand

3 3i and i

chapter 9: quadratic equations · 2017-02-12 · we previously have used factoring to solve...

Documents