Chapter 9: Quadratic Equations - 1.cdn.edl.io · PDF fileAnother technique for solving...

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

Chapter 9: Quadratic Equations - 1.cdn.edl.io · PDF fileAnother technique for solving...

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