x ioo(x...

Algebra 2 SOL Review Topic 1 Date: ___ _

Review Topic 1: Factoring and Solving Quadratics

7 Factoring Rules

Rule #1- GCF 1) 3a 2bc5 -9a3bc6 + 12a 2b3c6

3 a 2 h c � ( I - 3 ct c + lf '72

G )

Rule 112 - Difference of Perfect Squares 9 2 '

2) X -49y-

(}x+Z;)(3;< - 7j)

Rule #3 -Trinomial with 1 as a Leading Coefficient 4) a' +a-20

( 0t +s- ) ( v1 -11;

Rule #4 - Sum/Difference of Perfect Cubes 6)x3 -125y3

( X -5)) (x2

+fx_;

Rule 115 - Perfect Square Trinomials

8) 9x 2 +30x+25

Rule #6- Factoring by Grouping' b' ' b' 10) crx- -x +a-y- -y

' -� - � - y - -

(

\ x(a2_1r:, 2 )+�_ a,z-b;;:.)

(a'-b7j{x+y) C 0 --1- !:?)( ct-b) (x- I- 9) Rule #7 -Trinomial with Leading Coefficient >1 12) 3y2 + 5y + 2

(2 J- --i1/v1 + \._;; y , .,____ / lJ ./

What if it can't be factored?

) ' ? - '14 x- +

-;)Y-

Pr r,n C, /

)

3) 100x2 -100

IOO(x 2 -f) luo(x.+1)Cx-1)

5) 3x2- l8x +24

3(K 2 -e,x+i)'

3 (x-L/)(x-2-)

7) 64a3 + 8/

i('b!A 3

i C �°'-

9) 36y' - 84y + 49

(013 - 7 )2

11) 4x3 +4x2 -6x+6

� C ¼�>< 2. X -{ � )J( C,;v,-," o f b-e -A:;,.c for ed

13) 6x2 -Sx-6

r" I' J!' r=�" ·t-f-uz...r

(J';( -1- 2- ) (zx - J )

1

Algebra 2 Review Topic 1

Factoring Review How do I know which factoring rule to use?

Algebra ll

Cite:k fowGCF

- I

Trlrcmitcl

I

I;);iff.e,, -1i!YH;:l!-nf

��5

2 PeN:ctSq�

Tr'ln:aiicll.5

= I -R�:e: FOE{.

/ t.=:ing

Coo.fficNmt::: 1

Sumfr'r11duct Tcl:k

Factoring Mixed Practice

15) 2x2

+8x+3x+J2� \�

Jx()(±!/)+J(Xf )

[,,;:-fl/) (2x-r-3)

I

--1, T�rms

I

I-fu-'O!'..ipiq;

WOO.dirg C=tL • 2

I � & Ciu!:ck !;

16) 2x2

-7x-15

( 2X + 3 ) ( X - 5)

18) 3x2-18x +24

=?(x 2_ �x:+i)

s( x -4){x-2-)

2

Date: _ __ _Algebra 2 Review Topic 1

Solving Quadratics

What are the different ways to solve quadratics?

#1- Solve by Factoring

19) w2-8w-9=0

C a'lf ,'\ -oVv-'ijt_ I j -·

vv-q=o v,1+-f.::o

\;JC: -,

#2 - Solve by Graphing

22)

20) 25 p2 - 36 = 0 21) 3x2 = l 6x - 5

(':if-f&) (Sp-l )=o 3x: 2 -lfRxff: .::: o

Sp +G .:::o f;p-& =o (5x -/ )(x_ - s-):o

=-& sf�r,, Jx-/::o x-S=o p:::: - t f=- ; YX : J X = S-

? - � ;? \. S,; " )

x-in-ft'xC.t{)+ - foDf = sw lu_+iovi = z.e..rc,

X

#3 - Solve by Completing the Square

23) p2 -12p+36 = 0

p 2 - p +3 (p -. -3 I; +J �

[p-G,)2

� '2 � .t /0p='7 i_{,,?

-b±✓b2

-4ac#4 - Solve by the Quadratic Formula x = -----

2a

24) x2

-6x+2l=0 25) x+2x2

+1=-l-x

a.::::f lo=-& c:c:.2/

x :: - C-r,,) ::- 'Y fro) 2 - i.f Ct )(]. 1)

2(/)

X :c (,, ± t=1i X :: G,!: 77/p• 3,

x =

q-c::.2 b-=2 c =2 � r· --· ,�·-�-.·,·"=�'�"�- "-·�-t

X ::: - C 7_) ± 7...) 2. ... q' (<-) CL) .. -� ' "· "�" '•. e� , .. - . , •-· --� ·=-�•��

2(i; X-:: -2 .... ±l-r2

X= -2

-----

Algebra 2 Review Topic 1 Date: ___ �

How do I know which way to solve? Try to solve by factoring fi;;st, if you can't solve by factoring use �he quadratic formula or s?lve by

/ completing the square. I..A'ILCK. '1P7,<.,r- t?.,n.Swer //J r'tt., 1 nj u:,c-•1<- ,

G 'rap /,, {h.t '(_ IA..� d ro. /I 0 - It m 17'.. ,tq, S s ' ' Solving Quadratics Mixed Practice ,x .• 1 �'::;-c.e.p·fs )26)x2 +4x=3 27)2x2 +5x=3V2 +tfx-3=0 2X 2 1-S-X-3 =o

)( 2 _LJ_)C -l'f � 3 +Cf ('2-K - I) (x. -f .3) :o 1cx: f2) 2 ='fi 'Zx- I= 0 x'.-1-3-=o

Xf 0 :::+�.'r - )(:::-1=; � � X=-3 X::-2±1'7

28) x2

= "Sx

x 2 =o Y:[>C-S):o

0

30) 3x2 -5x+2=0

C, x -1- f) (x ·- ;) ) = 0

: .:::= 2

l-] 1 2,) Describe the Nature of the Roots of a Quadratic

A) 2 Real Rational RootsB) 1 Real Double Root C) 2 Real Irrational RootsD) 2 Imaginary Roots

29) x2 + 3x -40 = 0

Cx + 'l )Cx- >

31)

What are the nature of the roots of the following quadratics with roots?

' -oI -

32){±¾} 33){±¾i} 34){0,-S} ,

1 R eM R cd1 irt,) M ;). ! m °' 3 i'n a.r 'J 2 12. e»J. K a_h rm.J

35) {2 ± v'S} 36) 37) I � R.,e_,� ! r roA1 0n_,:,J :: \

' \ /;;_ \ /"--

'

4

Algebra 2 Review Topic 1 Name: __________ Date: _ __ _

TES

Factoring Examples: Rule 2 Difference of Perfect Squares

a' -b2= (a+b)(a-b)

8la1 -64x4 = (9a+8x2 )(9a-8x2)

Rule 3 Trinomial w/Leading Coejf=J

x2 -4x-32=(x-8)(x+4)

Rule 4 Sum/Difference of Perfect Cubes

(a'+ b3) =(a+ b)(a' -ab+ b')

or

(a3 -b3) =(a-b)(a2 +ab+b')

Rule 5 Perfect Square Trinomials

a2 + 2ab + b2

= (a+ b )2

a 2 -2ab+b2 =(a-b)'

9a' - 30ab + 25b2 = (3a -5b )2

Quadratic Example: To solve a quadratic equation you may be asked to find the solutions, zeros, or roots. These answers

will also be found on a graph (called a parabola) as x-intercepts.

Note: A quadratic equation can have two solutions, one solution (a double root-touches the x-axis and

turns around) or no real solutions (graph does not cross the x-axis).

Solving by Factoring:

1) Get the equation equal to zero. Move everything to left side.

2) Factor the left side using an appropriate technique we have learned.

3) Set each factor� 0 and solve.

EX) Solve for x: 6x2- x - I = 0

(2x -1 )(3x + 1) = 0

(2x-1)=0 (3x+l)=0

1 1 x=- x=--

2 3

5

Algebra 2 Review Topic 1 Name: U,�$ Date: ___ _

R R CT!CE

1. Factor completely: 25£ - 4A (5x- 2)' (Sx-i-2)('5x-2.) B. (5x + 2)2

© (5x + 2)(5x - 2)D. (25x + 2)(x - 2)

2 2. Factor completely: 9x + 24x + 168 (3x+4)2 (}x + L{ )

2

B. • (3x -4)1

C. (3x + 4)(3x - 4)D. 3(x + 2)(x - 2)

3. Facror completely: 27/ - IA (3y + 1)(3/ - 3y + 1)B. (3y + 1)(9/ - 3y + 1)C. (3y - 1)(3/ + 3y ·· I)@ (3y - 1)(9/ +, 3y + 1)

C3�- ()(q12 Ju I )

4. Factor completely: I25x' ) 64A. (5x + 4)3

B. (5x + 4) (5x2 - 20x + 4)C. (5x + 4)(25x' + 20x + 16)@ (5x + 4)(25x' - 20x + 16)(57C + 1 ) (}SY-

2 - + r ti,)

5. factor completely; 49x2

-- 25 A. (7x - 5)2 ( ?X 1 ( 7><.- S"

B. (7x + 5)2

8(7x+ 5)(7x- 5) D. 7(x + 5)(x - 5)

• 6 6. Factor completcJy: 8/ + z

A. (2y + z')' (}J +;;?;) (!j ,_ -· ¥ Z �)B. (2y + z')(4/ - 2yz + z2

)

C. (2y + z2)(2/ + 2yz' + 1)@(2y + z2)(4/ - 2yz' + z")

7. Factor completely: z' - I sz' + 8 lz{Ji z(z ·- 9) 2 � ( t; ::,.. - f g 7, -I 1: I}

B. z(z + 9)2

z1; ( -t;- _ q ) 2C. z(z+ 9)(z-9)

D. (z + 9)(z' - 9z + 9)

8. Factor completely: 4x' - 4000xA. (2x - 2000)(2x + 2000)B. 4(x � lO)(x2 - !Ox+ 100)

0 4x(x - l0)(x2 + lOx + 100)D. 4x(x + 10)(/ - !Ox+ 100)

,

tfx (x3 - /Do o)

t\ (x - ii>)(x:,.. +

6


R Use your graphing caicu!ator for questions 1--S. 1. Below is the graph of fix) = -x' + 4.

y

b!Jdjtltfill-+h;-;---;-cil!lc:-;l-:-llf---;cT;;---;:-t-.,_X

According to the graph, what are the solutions for -x2 + 4 = O? 0 x = -2andx = 2B._ x= -4andx= 4 C. x

= -2andx = 0D. x= -4andx = 02. What are the solutions for3x2

- l 4x ·· 24 = O?0 B.

x= -1andx= 63 x= -1andx= 03 C. x = -6andx= t, D. x = -6andx = 0

]K t--Lf:::::. o

3>< :-'-f -� '-f -· -

Name: ___________ Date: ____ _

}. Below is the graph of fix) = x2 + 14x + 45.

According to rhe graph, what are the solutions for x' + 14x + 45 = 02

G x = -9 andx = ···-sB. x

= -9andx= 0C. x = -5andx = 0D. x = Oonly4. Which is true of the solutions for2x' - 28x + 98 = O?

A. All real nwnbcrs are solutions for thisequaci.011-B. This equation has two distinct real@

solut:ions.This equation has orJy one distinctreal solution.D. This e.qua1ion has no distinct realsolutions, but le docs have complexsolutions. \ l. '

7


SOL Review Topic 2: Rational Expressions and Equations

Simplifying Rational Expressions Factor and Reduce!

2x2

+3.x-2

l)

2x2 -8 :z z__ If)

Multiplying and Dividing Rational Expressions

') I

,;:,..X-'

2 (x-z)

--, ', i• I,_){ - '

:.,. ---6x+ 12

5x f �I G l 2J /))?C -� -

(2.� --· -&"K / y,2- /

Adding and Subtracting Rational Expressions YOU NEED A COMMON DENOMINATOR WHENEVER YOU ARE ADDING OR SUBTRACTING FRACTIONS! Do not cancel on top and bottom! Get a common denominator & then add or subtract the numerators.

(x.+2--) 8 4 . LCD:[x.+z)Cx-2.) c+d d-c . Leo: loC aq2�---Cx-2-) s) -+--

, ___ _____.-

tx+zl,!-2) �+i)(x-z) - ·-, 2cd' 6c2d

'Dx+/L,-fx.J--�-r- _ t.fX+2-4 · (3c)(c+-d.),(9)(a1) = 3c. 2+3cd-cP{-1--d 2

C:<+2-)(x-2-) -Cx.1-2) (x-i) (3e, )(;;. td. '-J "'". Cr;,c.2 �)(r.i) :c '-d 2 I:·Solving Rational Equations /// 3 c L +- 2 cc( +d ' Multiply both sides of the eq

.·uationb\'.,the LCD then cance

·�· .

t

.

e fractions and simpl

J

if . '. .

. (:, .. . ,_ d·;- -

/' x + 1 = 5x +__l_ ---------)

(p (J.-2 J 7 _4 ___ 2_ = _u_ (v�'i-2-)&t--;; __ .. __ .. •

�(x-2) 6 x-2 u+2 u-2 u' -4 ' :!J 'H(-'ic.::; 0 ... • .. ·-...

:l(X·H)::: 5?' .(x-:i-)-/., C, ( x'= «.�\ 5 2 "? . L/(14.-2) - 2(v..+2):: IA :z-l{�-2 = 5x"-!OX f-/p ·--� '- e-- ) '-(11c-'Z - 2.-Vs. -t.f::: u._ ' x-z.:::o J

0::: GX 2 -17.X ./-- If � -- .... --· < "'- - (2 ::-: v.( 'Sx - 2 )( x - 2 ) = CJ FxcvJ�1.,Ji- "'- ::: 12-

Mixed Practice Simplifying and AdBingvand Subtracting Rational Expressions

l x+7 _ ' 4 i) (x.--z)(X-1-3) 9) x2

;-

7x+l2 x2

,+7x+l0\__x-2 r+x-6 x·-x-6 r+x-20

()(t-7)(K+-3)- 4 = Cx�z)(X (x�JJ(x--!fJ- ( Y: +-{ cc �

:?

/+ x-::-<e_00C-;;r-?;5&.;{'j.(;;c:;;(:i<:r4').

qx_ --= -23 -- 23 '

2-Sx 4x-5 10)--+-x-9 9-x

9

I

-s-x+2 Lfx-S: '�� -{- .-� x-q -11x�"f";· -2(x+J)

,_ -s x-+2 - C +><-..?J_�

'" ....... -·�-.. ··�-)( _# I

8

Algebra 2 Review Topic 2 Name: _ _________ Date: _ ___ _

T S

Simplifying Rational Expressions 1.) Factor the numerator and denominator

x2

+4x+3 Example:�,--

x· -x-12(x+l)(x+3) (x+l)(x-4)(x+3) (x-4)

Multiplying and Dividing Rational Expressions

2.) Divide out any common factors

1.) Factor all numerators and denominators (GCF, Unfoil (trinomia Is), Diff. of Squares,etc) 2.) Divide out common factprs (reduce). Example 1-Multiplying:

x+2 � x+2 � x Ex. - ---

x2 -4X x (x+2)(x-2) x-2

Example 2-Dividing: Change all division problems to multiplication- 'flip and multiply'! Then follow steps above.

X x+6 Ex.-----

x-4 x-4

X x-4

x-4 x+6

X

x+6

Adding/Subtracting with Unlike Denominators Multiply the numerator and denominator by what is missing from the factors of your LCD (also called the Least Common Denominator).

Ex x+l+� LCDis 6_(2)_x+l+�-(�)=3x+3+ 2x 3x+2x+3=5x+3

2 3 3 2 3 2 6 6 6 6

Complex Fractions Find the LCD for each set of fractions. Then flip and multiply.

l ,, 1 3x l + 3x-+� -+---Ex. _x __ = X X X

5 5 4x 5+4x -+4 -+-X X X

Rational Equations

X

1+3x X 1+3xX 5+4x 5+4x

Step 1: Multiply each term of the equation by the LCD. Step 2: List the values that must be excluded from the solution. Step 3: Solve for the variable. Check your solution in the ORIGINAL equation!

Ex.1-S __ _]_ = 01 LCD is (p · l)(p + 2) p-1 p+2

5 3 (p _., l)(p+ 2)- ----(p-l)(p '- 2) = 0(p-l)(p+ 2)

p..-<) p'+-_2

S(p + 2)-J(p-l) = 05p+l0-3p+3=0

2p=-13

9


RE PR ii E

2. \i'hich is equi·valent to a(a + l)1 . I5f(b + 1);,3b a+ 1 ·

A Sob_ · a+- 1

B. Sab(a + I)b+l

@sab(a + l)(b + I)

D. Sab(a + l)2(b + 1)

3. Which is equivalent to d + b + _a_, · a-b a+b·

A.2a+ b

2a

B.i - 2ab -+· b2

(a+ b)(a - b)

C.2a'-3ab+ !l

@ (a+ b)(a- b) 2;} +ab+ b2

(a+ b)(a - b) b -·- 1

4. \\7hich is equivalent to -f '

(!j b(b - I)if:)

B. b2

(b - 1)

® �(ll+J,1 CC\-h) ((it -\c.,X@,;,-J (q, b) Ca-s,)

b,-1 - ,

+b)

Date: ___ _

6. Which is eq uiv:ilenr co5c',f' '\

.1oi+20J+rnd',-::> IO (e, 2 �2c,:/-1-d "- )3cd(c - d) if ·

' 7,

2:+U /t.

cd' 3(c + ,t/ 0

B.a/2

3(c + d)(, - d)

C. 3(/i2

df D. at'

3(c - d)

2-�Which is equivalent to --2

, x- .?£ 4

A 8-x c. 4-x

-�-- --xx-·

@ 8-lx D. 2 ·· 4r �

8. Which is equivalent to the expressionbelow?

-1._ + -"'-- - 4 x + 1 x - 1 (x + l)(x - !)

C. 3x1 T x -· 7(x+ J)(x-1)

,::-,. x'+4x-7 \_V (x+ l)(x-1)

c.. d �

3 (c-d}'

t./ y:--;;_ z.

• '

' 4:r 4

I'-.

tf

lf-X l,r2 2 ('f-X) i-2.x--;;~ • --

7 6)<. 3X.

(1) . --1...Gc-,).f--- __ X_Cx+IJ( X+/)&-1) <?'-/) lX+I) --

:?,x- 3 + )(2- +-_t!, .-..'----Cx-1-1 )Cx-1)

-3>(

Cx+()Cx-1)


RE R Tl E

I. Solve the equation for n:

"--"-

:

h 2 -I--Z;i":::[ (,, + z,,5 = 10) hA. n = -sonly

@ n = 5on.ly

h 2 - 10,; 1--2':; :::: 0

r '2

1.._,-,-s; I =o

C. n= -5orn�0 5 h ""' S°

D. There iS no solution.

2. Solve the equation for x:/ 4r,z_ 3x\)(-·.2 · _x-2 -x·�;

A. -x = 2 only l.{ X - ;J._ -:::. 3 )(

B. x= -2only

C. x= 2orx = ·-2(0 There is no solution.

D. c. = 15

11

.::: 0

Algebra 2 Review Topic 3 Date: ___ _

SOL Review Topic 3: Simplifying Radicals and Complex Numbers

Simplifying Radicals

1}✓200a2b 2c 11 2)&/lr;;/0-0 2 z l 2 e,fD ifff.,; .. � 0. '1/7 ' ·C

!O(�b/c.5 12c ' .

Operations with Radicals

s)\/nx3y .Jsoxy3 6} (2- ✓3)(2+ ✓3)

---,-::--�-:,-·--, v30•2 · .2s-2 · x'-1,y !f l-{-1q'

G>•5,2·x 2 · "

GO X 2

• J ;;a

9)-✓ih'-W

Trf·3')<<f·X ·-/lf-2·X11·X Q.i'l§x - ;}. x 12.x

Rational Exponents

Express the following in exponential form: --.~- f 11)¾(3x)2 lfi< :JJx ? 12) ✓7 � 7 .2: ·

;;_ I .,.

Ox )3 ==-3 :ix·i

Express the following in radical form: Jr::_1 I I \Jr"

15) r 3 :::

r} = r{]r =' �:, ' i!r; r

Powers of i 1s} 1= TT

j5 = (.,

, " - (.,

. 1,

. 19) i'' � L 32- -= / -rtvu-e-FDr�

Imaginary Numbers

21)H◊

,Y-,2.r;. 2.

� i, Ti

i2 = C 1-1 }

'--=-I i'=i

2·c=-Z

·6 I l = - ., Ll = -

20) i 102 = - / i roo = I -tC,.e«.fyrl..,

'10/ • ( . = C

22JH·H

7- / • 3.' • 1- I· 3 7

't'l/s·i:D� "' I°/

-r · s - -- 3

4) �-8a4b9c 11

� I · ,g- •o, J ,a_ ·b c1.c "·Z"- ;/.O. /.:,

3c

3 'Q OI..C :;c

7) £4 - .!/81 + efj�-if:;7,3 + V3 2V/T- 31/3-1-�

0

10) 2 c�+-rz:)

(3-F2)C3t,-i)

t2+:2,12 C, -1-?.ii- �-,.,�·

Cf -2 1

14) �( 4a)'f,

( 're, )'f

.4 fl =

.3 l = I

_2_ (2,3i,) 23)( 2- 3i)(2-1-;s i)

Lf +- h 2 y f G;_ 4- q&2. :' 'ftC, 12

;.i.,,�; t (, tC V


, Lb'✓/) do I rOperations withjomplex Numbers 24) (2+i)(4-3i)

6 -&i +t..fi-'3i 2

[-2i: +3 I '

'

I I- 2v

25) (1- Si ✓3) - ( 4 + i ✓3) + i

-3 - r( ri -1- �

Solving Radical and Rational Exponent Equations

26) (Sa-5) 3 +I= 3,? I \ ,

l(i - v r . ," ! '/

\ I;, Ge _c ) " 1:::: i 2 ' \.'- J ) �/

)bl-) :c g

)v\ = 13

[>, ::: 2

27) .J X - ] + 2 = -]

l'x-1- = - 3� I f'Jo )O I Oh

.1. 7 - -11

s><-- -

I s=- >< -== c;;

x_=-3 (J

Mixed Practice with Radicals and Complex Numbers SIMPLIFY OR SOLVE: l -/ -l (

29) -2�4x-12+1=-3 \.fr:-•

-2 "v 'fx-12 :::c -'f

{:� 'hc-12)u

':: (z 14

Lfx- /2 :: /(, 4x== x=I

32)�-.µ

cri• tl1

"i2M

35) (2+.Js)(J-2.Js)

Co - lf''r: + 1 5 .... ') ,r;;.I 1;, - ,,,.. , "··>

& _,.,a 1 -·- l

30)x2 + 49= 0

)( 2 = _if q

1-2 "' +

33) 4�16p4q9

4•Zlplq1

�

o\r\�4�

Ti-

31) ;'3

i. q 2-= f t' 9 3 = i

x.-2-=- ! l,

v- 16 13 /"',.,... ! -0

Algebra 2 Review Topic 3 Name: _________ �Date: ____ _

PLES:

Simplifying Radicals To simplify, use � = % • '!fb or break out into prime factors looking for the same repeated factors (2 or 3 or 4 of a kind-depending on the index).

Ex)132x5y

4z 19 = 116-2-x4

• x- y4 • z 19

· z3 = 2xyz4 12x · y4z3

Simplifying Radical Expressions:

Ex)� +8x+ifi7 -16x = 2ef; + 2xef;-8x = 4xef;-8x V8x + 8x+ JJ16x4 -16x

Multiplying: ef; .'[;=!!]-;;;Simplify each first, then multiply. Ex)2✓12 3✓3 = 6,/36 = 36

Dividing Radicals: Answers cannot have radicals in the denominator. We need to rationalize the answer by multiplying the top and bottom by a radical that will 'lift' the root sign. If the denominator is a binomial you must multiply top/bottom by its conjugate.

Ex)�=�. JSx 5xJSx JSx JSx JSx JSx 5x

Radicals or i's in the denominator: Multiply the numerator and the denominator by the conjugate.

Ex)�=�_ 5-i = 25-5: = 25-Si

5 + l 5 + l 5 - l 25 - C 26

Adding/Subtracting Radicals: Simplify then combine like terms. Ex)2✓!2 -3✓3 = 2 2;/i-2✓3 = ✓3

a Rational Exponents: Divide the exponents by the index: x

b = 'ef7iExamples of rewriting:

Ex)) =IVI

Complex Numbers a+ bi Remember: If there is a negative(-) under an even root, pull it out as in i. Ex)Cycle of powers of i: I/ = i, i2 = -1, i3 = -i, i4 = 1 I

14


EXTR PR Tl E E:

1. Which i◊ equivalent to J?A \/?

0%':, r ···c. \'61

D. 33

2. Which is equivak:.nr to 1/23?

A 25

l3. 5½@ s; D. 51 5

' 3, Express x3 in simplest radical form. G /[,/ l �

ll. ,!c¼x> 1 3 · X

C. VJ,,, x· D. X\/X

4. What is the difference?(18 + 11,) · (20 + lli)

@-2B. -2 � i

C. -2 - lliD. -_-2 + 22i

-· 2.

2. Which i.s equinlent to (8 + 5i)(8 - 5i)?A. 39 G,"f' - Z5 � -iB. 64 - 25iC. 64 + 25i

@> 89

5. 'Which is equivalent to 'V27n12 ?(;y !. ' -1..

V 3:n J.7 b h B. JiiC. 32i D. 27'fnJ6

C.5/5 (A x1xy

l

V' '

D. J

X XJ

7. Which is eguivalcnrm (I6a2)1? ·A 12VJ ? •

{r,/f •(n . B. I2awi "' c. sVl 'b. o_ �

z

'· -'·

@sa..ra t.i 1)'!,\'{•a_

8. "W11ac is rhe difference, .in st-andn.r<l form?

D. 13 - 5i'5

6. "'w'hich is equivalent to (9 - i../5)(2 + i{S)'A.B. 23-7;../5 /'6 f qi,f5 - 2lT5 .. (2.1Js

l3+?i../5 1i t7r,f5 +r; . 34 - i9i

D. 34 - 19i../5 2 3 f I i''(s'

15


E TR PR Tl E F:

1.

2.

3.

3X+/ 3

r -2, , 2 Solve for X\_✓3x + V7'.'._4 J

::::. / '9 = fS

x=�

A. x = 1 C. X = 6{Dx = 5 D. x = 16

Solve for d_ 1/a+ 3J:V) 3 C( -/- 5 " S?"\. qtc:: ,;-A. a=! �a=)

B. a = 3 D. a = 11

The functions y = "Ix - 3 andy = -3x + 13 are graphed below.

Use the graph to solve1/x-3 = -3x+.!3forx.A. x = O

1J y:._--- I

l:I. x- = IC. x=3

x = 4

4.

5.

Angel correctly graphed d,e funcrionsJ = ✓x+4andy =3x-12andfound

) 2 their point of intersection on his c3\cnlatoL1his is how his screen looked:

Based on Angel's results, what v"1ue of xisa soh,rion for ✓x + 4 = 3x - 12?A. x = O

B. x = 3 0

x =

5

�. There is no solution.

Solve for n: Y3n +} = t

(>, n =l t'.) 9 B. n = t

x =�

C. n = 9 Jn

D. "There is no solution.

r 7 -: .!.... h - -

16


SOL Review Topic 4: Solving Equations and Inequalities Absolute Value Equations, Rational Equations, Radical Equations, Quadratics, Systems, Inequalities

Absolute Value Equations

1) 312x-11+2=11 2) l9+4xl=5x+l8Cf +lf-X.-=-!;X+ if> or q +Lf-x =-(Sxf- /'l,)-1 = X or '1 =-Sx -l'l

3 (2-,:-7 I = 1c;

I 2)<-7 I :: 0}x ::5 or Zx- -::::-t;

'f)(:.:::: -27 X"'-3 /

)(:::C (;, OY "A= I

4) lx-31-2::::1[x-i/:s3

X-3 :=: 5 ,JJ,i� X-5 ► -3x :=, C, !?Ad L 0

Radical and Rational Exponent Equations

5) 2=�✓5x-l+l3

I /

JC I = t

Rational Equations

yf l / 5 7) 1---1::=-/3 9x/3 18

)

(_1 ; "!V' -' 'I'-

r- ·,)

r'i;){ 0 '

-- '1 -L --

/

I

6) -2=2(x-l)3 +4

-ro:: (x-1) 1

r

.,.,J ,, ! \3 --2. J- ,I ( /)' '; )1

..,/,/=' ( x- .;, 'C: '

* Uili-i<f ,-:q,,/ r �! f' f;X !J /f.'•!/

"!JC/ •

A_ +f) U<-/ )

8) ----,-=0'}'

\'._ x+l x· -1

J(x-1) - >X.:::. c,2x -s --S'X :::c,/ -

-2-X -3 ::: 0

-2-x. :::: 5

b r €.-v'-i O {�. Less Tf,f A /'Jd,

·<-z

Algebra Z Review Topic 4

19) 21x - 212: 34

[x:-2.1>-lf > tf or-

� or

2 -l{

·x; =- -2,

;}_ I ntt:j l Fl IN'.}

/{ao(5

X :: .. -· '- I

14) x2 -6x = 1

X 2-(n<.-i

2_fc,x., +1

n:-s) ;_

- I '- f • J

--1'-15-- Li

Y.-.3 :::c ± 7'io' k '° �i

' f\

;;).._ f1,u) { rn:L.J7 cr/L,J/

�<l"v·fs

18) 4��x-1 =-84

I -='\ ? 3

t��x -I /::(- 2)

I - 1 - -',? ,t X-1 -

21) 5x2

- l Ox+ S = 0

'5(x2 -2x. +! )=o

= I

18

Algebra 2 Review Topic 4 Name: __________ Date: ___ _ _

X LE :

Solving Absolute Value Equations: Set up 2 cases and solve for both: Ex] l2x-

5I = 3

- ------------�

2x-5 = 3 2x-5=-3 2x=8 x=4

2x = 2 CHECK your solutions!

x= 1

Solving Absolute Value Inequalities LessthAND (:s;, <) (graph is 'in between') versus GreatOR (?c, >) (graph opposite directions)

Set up 2 cases-Remember to flip the sign when setting up the 2"' negative case! Ex] 1: 12x- s1 < 3 Ex] 2: l2x-5I > 32x-5 <3 2x<8 x<4

Solving Quadratic Equations:

and

l<x<4

2x-5>-3 2x>2 x>l

2x-5>3 2x>8 x>4 or

2x-5<-3 2x<2

X <)

To solve a quadratic equation you may be asked to find the solutions, zeros, or roots. These answers will also be found on a graph (called a parabola) as x-intercepts. Note: A quadratic equation can have two solutions, one solution (a double root-touches the x-axis and turns around) or no real solutions (graph does not cross the x-axis). Solving by Factoring:

1) Get the equation equal to zero. Move everything to left side.2) Factor the left side using an appropriate technique we have learned.3) Set each factor� 0 and solve.

Ex] Solve for x: x2

- 3x - 10 = 0(x- S)(x + 2) = 0x-5=0 x+2=0

lx=-21

-b± ✓b2

-4ac Solving using the Quadratic Formula: x = ------

2a

Get the equation equal to zero. Move everything to left side. Find a, b, c and plug into the formula: Don't forget to simpllfy!

Ex]

Solve for x: 3x2 - Sx + 9 = 0

-(-5]± ✓(-5

]2 -4(3](9) = --'-'----'-'---'-----'-'---'-2(3)

= 5

±� =15

±i✓83 1 6 6

19

Algebra 2 Review Topic 4 Name: __________ Date: ___ _

T

Rational Equations

Step 1: Multiply each term of the equation by the LCD. Step 2: List the values that must be excluded from the solution. These are values of the variable that make the denominator� 0 (these values make the equation undefined). Step 3: Solve for the variable. Check your solution in the ORIGINAL equation!

Ex) 5 3 ---=0

3a 4a2

LCD is l 2a2

l 2a2 _ ( 20 _ 3

' = o) ] Cl 4a·

20a-3 = 0 20a=3

Ct=-

20

Radical Equations Steps to solving radical equations: (1) get the radical on the side by itself ("isolate the radical") (2) square ( or cube, etc) bothsides of the equation (3) solve for the variable ( 4) check for extraneous solutionsEx) -2✓3x+l =-10

✓3x+I = 5

(.J3x+l)' =(5)2

3x+1=25 x=8

Non-linear systems Ex) Solve linear quadratic system: y � x' · 4x · 2 and y � x-2

1. Enter the first equation into Y,.2. Enter the second equation into Y2 .

3. Hit GRAPH.4. Use the INTERSECT option twice to find the two locationswhere the graphs intersect (the answers).

2nd TRACE (CALC) #5 intersect

Move spider close to the intersection.

Hit ENTER 3 times. 5. Answer: (5,3) and (0,-2)

I

I

,... ...... ---

--

Iratu·s.;:.:ti,:,r1 :-:=E: -

I /'

_( __ .... --,.,f

/ '

' I . ._ __ .

11'=3

F'l❖ti f'1❖�;;: F"l❖t3

,s,s:,,-4x-2 ·-..Y;;:EIX-2 -... 1./J=I ,,l/ lj= -...v�='·.\-'ti= ',l/7=

20

Algebra 2 Review Topic 4 Name: ___________ Date: ____ _

PR TICE :

1. Whal is the solution s�t for j2x -r 31 = 9? 4.

j / ll \,,· _

,...,

i :.:::; t "'- ,?..r

- ,.,,� -' ( --- LY-'1 ()y -2 ::-x-'-f

x:::z

A. {-9, 9}B. (-·6, -·3J@{-·G,3}D. {"·3, 6}

2 x:-t--3-=- q or /v-1--J!:::-!42y:_:: G oi- 2r:. -::: - Z

What i.s the solutiou set for [x-21+4=2x' @12}

B. {4}

C. {2, 4}D. {3, 5}

2. To solve [2x -- 2[ = 6, Navi graphedy = [2x - 2[ and y = 6, as shown below.y

Wha1. are the so.lutlons for this eg_uacion? A. x;;;: -Gorx= 6

(M x '" -1 or x = 4 '�.,/ C. x = -2 or x = 6

D. x = 4orx = 6

2. What are tbe solutions for3x2 - 14x -- 24 = O?0 x = -�andx = 6

B. x= -�andx = 0

C. x = -6 and x = 1 b . 3

,,-/'.. D. x= -6wdx_,.=--O '6/ 3)(i.-14x-2-f::.. 0

5. To solve [xi = -2, Elk graphed y = Ix[and y = -2 on her calculator.y-axis y = [x I

1 ! ��--- ---.r-- -\ ::t

1,: ..

1c�=='-'-".i.::iSJ!-<\, · ' ! -x-axis

y= -2 'w'1iat are the solutions for this absolute value equation? A. The. solurio.o i� x = -2.B. 1he solution .is x = 0.C. The solution h x = '-2 or x = 0.� There is no solution because the

V graphs do not intersect.

4. Which is true of the solutions tor2x2 - 28x + 98 °0 O?A. All real numbers are solmjons for dllsequation.B. Thls eq ua:tion has wo CUStinct real

G scludons.'This eqU:ation has only or1e distinnreal solution.

D. Tbls equation has no di.5tinct realsolntions, but it does have complexsolutions.

0,i (,Z - <;;

3 x +Lf D

't =

l \

21

(

Algebra 2 Review Topic 4 2_ . Name: __________ Date: ____ _c;,- X -!.f 'x + L/ = q -1-'-f

R T ICE H /1 ,_fr:< ··2)' :::: �,,-( I? .73/ ,; ';; .. ,111· / 3. Solve by compleOJ.1.gti1e square: · �� L -- ' / . t,":-;; 7. Solve: x' + 2, = 10

4.

x' - 4x = 9 )(-=c ']_ :f:: I [ jA. X = 4 :±: ,1 J3

@x = 2:±:fil C. x = 2±: ✓5

D. x•• -1 andx = 5

Solve using the quadratic formula.: 2x2 +4x-3=0

lJ_ X = -4 :±: 10·C. X = -·1 :±: 2·.{[Q

®

A. X = 1 :±: -y'ffB. x = 1 :!: --ffo

= -1 :±: --[ITr;···-= -1 ±: '110

ut::: L �-:: ··<l C'- 1 8. Solve: 2x2 - 8x + 9 = 0Gx = 2±:1:

B. x = 2 :±: 2Ni

C. x = 2 :!: 8iD. x = 4 :±: 1'(

X =.

'i5_±Zifz� ---·

\.4. Angel correctly graphed the functions \ y1 = ✓x + 4 andy, = 3x - 12 and found\.

their point of intersection on his calcn!amr. \ 1Lis is how his screen looked: \ /q) 2 )< 2 + lf Y:. -3 :::: (J� \Y 2 +: "', Q-2_ I _u. (..'.:-J + j ,- i:;,- I _____ _

J •-t"+l ';( ::: -_ CL{) ± rrc -; ;=�- Z) (-3)

((

Based on Angel's results, what value. of xis a solution fur ✓x + 4 = 3x - 12? A. x = 0

i) ::-2irlo

2

5.

Ax=

3

V x=

5D. 1bcre i.<; no soltu:ion.

Solve for n: ✓3n + ½ = ½

D. '1hcre is no solution.

I Jn + 'f :.: (

2. Solve this system of equations by the substitution method:

'Which is rhe solution set for this ;( ?. � 75)(

nonlinear system? A. (2, 0)

(v' (2, 0), (6, 8) c. (0, 2), (8, 6) \; 'l D. This system .has no solution.] ".. /. j

, , ,. '6 C& 1 r)

1- '[E2-

=o

.::.0 l

22

x ioo(x...

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