Name
Date
Class
Reteach
LESSON
9-6
Solving Quadratic Equations by Factoring
Quadratic Equations can be solved by factoring and using the Zero Product Property.
If the product of two quantities equals zero, at least one of the quantities must equal zero.
If x y 0, then
x0
or
If x 3 x 2 0, then
x30
y0
Use the Zero Product Property to solve
x ⫹ 8 x ⫺ 5 ⫽ 0. Check your answer.
Check:
x 8
x 8 x 5 0
x80
x 8 x 5 0
x50
or
x20
or
8 8
5 5
x 8
x5
8 8 8 5 0
0 13 0
0
0✓
x5
x 8 x 5 0
5 8 5 5 0
13 0 0
0
0✓
Use the Zero Product Property to solve each equation by filling in the boxes below.
Then find the solutions. Check your answer.
1. x 6 x 3 0
x⫺6
0 or
x⫺3
2. x 8 x 5 0
x⫹8
0
0 or
x⫺5
⫺8; 5
6; 3
3. 3x x 7 0
3x
0 or
0
4. 2x 3 x 9 0
x⫺7
2x ⫺ 3
0
0
2
5. 5x 1 x 2 0
6. x 4 2 x 0
1 ; ⫺2
__
⫺4; 2
5
a107c09-6_rt.indd 46
x⫹9
3 ; ⫺9
__
0; 7
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
0 or
46
Holt Algebra 1
12/26/05 8:41:51 AM
Process Black
Name
LESSON
9-6
Date
Class
Reteach
Solving Quadratic Equations by Factoring (continued)
Sometimes you need to factor before using the Zero Product Property.
Solve x ⫹ 4x ⫺ 5 ⫽ 0 by factoring.
2
Check:
x 4x 5 0
x 5
2
x 5 x 1 0
2
x 4x 5 0
5 2 4 5 5
0
x10
25 20 5
0
5 5
1 1
0
x 5
x1
x50
or
0✓
x1
x 2 4x 5 0
1 2 4 1 5
0
145
0
0
Solve 3x 2 ⫺ 12x ⫹ 12 ⫽ 0 by factoring.
Check:
x2
2
3x 12x 12 0
2
3 x 4x 4 0
2
3x 12x 12 0
3 x 2 x 2 0
30
or
0✓
3 2 2 12 2 12
0
3 4 24 12
0
12 24 12
0
x20
0
2 2
0✓
x2
Solve each quadratic equation by factoring.
2
7. x x 12 0
8. x 2 10x 25 0
⫺4; 3
10. x 49 0
2
⫺7, 7
13. x 2 10x 21 0
⫺3; ⫺7
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
a107c09-6_rt.indd 47
9. x 2 7x 8 0
⫺5
11. 4x 25x 0
2
0; ⫺6.25
14. 4 x 2 0
⫺2; 2
47
⫺8, 1
12. 5x 15x 50 0
2
5; ⫺2
15. 3x 2 6x 9 0
⫺1; 3
Holt Algebra 1
12/26/05 8:41:52 AM
Process Black
*À>V̈ViÊ
3OLVING1UADRATIC%QUATIONSBY&ACTORING
--"
™‡È
#OMPLETE)FABTHEN
AÊÊä
BÊÊä
OR
*À>V̈ViÊ
3OLVING1UADRATIC%QUATIONSBY&ACTORING
--"
™‡È
5SETHE:ERO0RODUCT0ROPERTYTOSOLVEEACHEQUATION#HECKYOURANSWERS
SXDSXD
5SETHE:ERO0RODUCT0ROPERTYTOSOLVEEACHEQUATION
#HECKYOURANSWERS
SXDSXD
X
X Ç SXDSXD
OR X
OR
X Ó X X x SXDSXD
OR
X
OR
S XÊÊx D
äÆÊÎ
X
x
™ X Î
XÊʙ D
X ™ XÊÊ£ D
X
OR
S XÊÊ£ D
X OR
X £
X X
S
OR
S
OR
S
XÊÊÎ D
X Î XÊÊÎ DS XÊÊx D
XÊÎ D OR S XÊÊx D
X Î OR
X x
S
Ó
XX
XX
xÆÊ£
Î
Ó
Ó
Ó
X X
xÊ
£ÆÊÊÊÚÚ
È
ÓÊÆÊÊÊÚÚ
£Ê
ÊÊÚÚ
x Ó
X X
£
ÇÆÊÊÊÚÚÊ
ÚÚ
Ê{ÊÆÊÊÚÚ
Ê£Ê
{
X X
XX
ÊÚÚ
Ê£ÊÆÊÎ
Ó
ÈÆÊÓ
XX
XX
£ÊÆÊÊÊÚÚ
£Ê
ÊÚÚ
nÆÊx
XX
ÇÆÊÓ
X X
ÊÚÚ
Ê£ÊÆÊÓ
Î
xÆÊÇ
XX
ÚÚ
ÊxÊÆÊ£
X X
£ÆÊ£ä
XX
£ÎÆÊx
X X
ÈÆÊÓ
£ÆÊ££
XX
äÆÊx
4HEHEIGHTOFAFLAREFIREDFROMTHEDECKOFASHIP
INDISTRESSCANBEMODELEDBYHTT
WHEREHISTHEHEIGHTOFTHEFLAREABOVEWATERANDTIS
THETIMEINSECONDS&INDTHETIMEITTAKESTHEFLARETO
HITTHEWATER
!RELIEFPACKAGEISRELEASEDFROMAHELICOPTERATFEET4HEHEIGHT
OFTHEPACKAGECANBEMODELEDBYHTWHEREHISTHE
HEIGHTOFTHEPACKAGEINFEETANDTISTHETIMEINSECONDS4HEPILOTWANTS
TOKNOWHOWLONGITWILLTAKEFORTHEPACKAGETOHITTHEGROUND
ÇÊÃiVœ˜`Ã
Ó
£ÈÊT ÊÊ£ÈääÊÊä
£äÊÃiVœ˜`Ã
A 7RITETHEEQUATION
B 3OLVETHEEQUATION
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
œÌʏ}iLÀ>Ê£
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
*À>V̈ViÊ
3OLVING1UADRATIC%QUATIONSBY&ACTORING
,%33/.
™‡È
œÌʏ}iLÀ>Ê£
,iÌi>V…
3OLVING1UADRATIC%QUATIONSBY&ACTORING
5SETHE:ERO0RODUCT0ROPERTYTOSOLVEEACHEQUATION#HECKYOURANSWER
1UADRATIC%QUATIONSCANBESOLVEDBYFACTORINGANDUSINGTHE:ERO0RODUCT0ROPERTY
SXDSXD
)FTHEPRODUCTOFTWOQUANTITIESEQUALSZEROATLEASTONEOFTHEQUANTITIESMUSTEQUALZERO
XSXD
ÈÆÊ£
£Ê
ÚÚ
ÊxÊÆÊÊÊÚÚ
Ó
X X
äÆÊ£ä
ÎÆÊx
£nÆÊ£
{ÊÆÊ£
ÊÚÚ
Î
£ÊÆÊx
ÊÊÚÚ
Ó
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
SDSD
X
X
SDSD
SDSD
SXDSXD
ÊÚÚ
ÊÎÊ
Ó
X OR X X X
™ÊÆÊÊÊÚÚ
{Ê
ÊÚÚ
Ó
XSXD
Î
XX
✓
5SETHE:ERO0RODUCT0ROPERTYTOSOLVEEACHEQUATIONBYFILLINGINTHEBOXESBELOW
4HENFINDTHESOLUTIONS#HECKYOURANSWER
XX
✓
X
S XDSXD
ÇÆÊ{
4HEHEIGHTOFAROCKTHROWNOFFACLIFFCANBEMODELEDBY
HTTWHEREHISHEIGHTINFEETANDTISTIME
INSECONDS(OWLONGDOESITTAKETHEROCKTOREACHTHE
GROUND
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
X
SDSD
X X
XX
ÊÚÚ
ÊÓxÊÆÊ£
OR
SXDSXD
ÓÆÊn
X
X X
X ÇÊ
ÊÚÚ
ÊÇÊÆÊÊÊÚÚ
Î Î
#HECK
{ÆÊn
XX
XX
£Ê
ÊÚÚ
ÊÎÊÆÊÊÊÊÚÚ
{ Ó
Ó£ÆÊÓ
X
X X
XX
{ÆÊx
nÆÊÈ
X X
X X
ÈÊ
ÊÚÚ
ÊÎÊÆÊÊÊÊÚÚ
{ x
nÆÊÎ
XX
XORX
SXDSXD
X X
X X
XX
™ÆÊ£
XX
£ÓÆÊÎ
XORY
5SETHE:ERO0RODUCT0ROPERTYTOSOLVE
TXETXE#HECKYOURANSWER
X X
XX
)FSXDSXDTHEN
{
3OLVEEACHQUADRATICEQUATIONBYFACTORING#HECKYOURANSWER
)FSXDSYDTHEN
SXDSXD
™ÆÊÎ
äÆÊÇ
SXDSXD
AK4up.indd 85
£ÆÊÈ
X X
X X
X X
X
™‡È
X X
ÎÆÊ£
ÎÆÊn
SXDS
--"
X X
X X
£ÊÆÊÈ
ÊÊÚÚ
Ó
XX
X X
XÊÊx D
X
X
OR X
X Ó OR
X ™
SXDSXD
XX
&ACTOREACHQUADRATICEXPRESSION4HENUSETHE:ERO0RODUCT
0ROPERTYTOSOLVETHEEQUATION
XS
3OLVEEACHQUADRATICEQUATIONBYFACTORING
Î
XX
SXDSXD
X X x ÓÆÊ{
ÚÚ
Ê{ÊÆÊÎ
OR
OR
SXDSXD
X
X £
OR
OR
SXDSXD
ÓÆÊÈ
X
X £ ÓÆÊÎ
X
œÌʏ}iLÀ>Ê£
OR X ??
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
85
X OR X SXDSXD
SXDSXD
Ó°xÊÃiVœ˜`Ã
SXDSXD
X OR X ??
SXDSXD
(OLT!LGEBRA
Holt Algebra 1
12/29/05 2:15:30 PM
,%33/.
™‡È
,iÌi>V…
3OLVING1UADRATIC%QUATIONSBY&ACTORINGCONTINUED
™‡È
3OMETIMESYOUNEEDTOFACTORBEFOREUSINGTHE:ERO0RODUCT0ROPERTY
3OLVEX XBYFACTORING
)N#HAPTERYOULEARNEDMETHODSFORFACTORINGPOLYNOMIALSTHATARENOT
QUADRATIC"YUSINGTHESEMETHODSANDBYEXTENDINGTHE:ERO0RODUCT
0ROPERTYTOMORETHANTWOFACTORSYOUCANSOLVESOMEHIGHERDEGREE
POLYNOMIALEQUATIONS(ERESANEXAMPLE
#HECK
X X
X
SXDSXD
XX
✓
3OLVEXXXBYFACTORING
SDSD
X
X
OR
X
X
X
…>i˜}i
3OLVING0OLYNOMIAL%QUATIONSBY&ACTORING
--"
XX
SDSD
✓
XXX
7RITETHEEQUATIONINSTANDARDFORM
SX XDSXD
'ROUPTERMS
XSXDSXD
&ACTOROUTTHE'#&OFEACHGROUP
SXDSXD
&ACTOROUTTHECOMMONBINOMIAL
SXDSXDSXD
&ACTORTHEDIFFERENCEOFTWOSQUARES
XORXORX
5SETHE:ERO0RODUCT0ROPERTY
X??
ORXORX
3OLVEEACHEQUATION
#HECK
XXX
XXX
XXX
??
??
SD SD SD SDSDSD ??
???
???
3OLVEX XBYFACTORING
#HECK
X X
SX XD
X X
SXDSXD
SDSD
SD
X
OR
p
S D
X
S D
S D
3OLVEEACHPOLYNOMIALEQUATIONBYFACTORING
X X X
✓
X X X
X X X
X
3OLVEEACHQUADRATICEQUATIONBYFACTORING
X X
X X
--"
™‡È
(OLT!LGEBRA
ä]Ê{]ÊÈ
--"
™‡È
(EIGHTFT
4IMES
4HEHYPOTENUSEOFTHETRIANGLEIS
FEETLONGERTHANONELEGOFTHE
TRIANGLEANDFEETLONGERTHANTHE
OTHERLEG5SETHE0YTHAGOREAN
4HEOREMTOFINDTHEDIMENSIONSOFTHE
FOOTPRINTOFTHEBUILDING
!LGEBRAIC$EFINITION
&ORALLREALNUMBERSAANDB
IFAqBTHENAORB
:ERO0RODUCT
0ROPERTY
£°ÓxÊÃiVœ˜`Ã
%XAMPLES
)FMTHENM
SXDSXD
4HEAREAOFTHESTOREISTOBE
SQUAREMETERS)FTHEDEPTHISGIVEN
AS???
X???
XWHATISTHEVALUEOFX
! # " $ ' METERS
* METERS
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
ÞiÃÆÊLiV>ÕÃiÊäÊÊäÊÊä
iV>ÕÃiÊޜÕʘii`ÊÌܜʜÀʓœÀiʵÕ>˜ÌˆÌˆiÃÊLiˆ˜}ʓՏ̈«ˆi`
,OOKATTHESECONDEXAMPLEINTHECONCEPTMAP)FXWHAT
MUSTBETHEVALUEOFX7HY
Ó]ÊLiV>ÕÃiÊÓÊÊÓÊÊä°
'IVEANALGEBRAICEXAMPLEOFTHE:ERO0RODUCT0ROPERTYWITHTHREEFACTORS
vÊABCÊÊä]Ê̅i˜ÊAÊÊä]ʜÀÊBÊÊä]ʜÀÊCÊÊä°
3OLVEEACHQUADRATICEQUATIONBYUSINGTHE:ERO0RODUCT0ROPERTY
7HATISTHEDEPTHOFTHESTOREINMETERS
& ( ' ,OOKATTHEALGEBRAICDEFINITIONINTHECONCEPTMAP$OYOUTHINKAANDB
CANBOTHBEZEROATTHESAMETIME%XPLAIN
4HEPARKINGLOTWILLHAVEANAREAOF
SQUAREMETERS4HESHORTERBASEISM
LONGERTHANTHEHEIGHTOFTHETRAPEZOID
ANDTHELONGERBASEISMLONGERTHAN
THEHEIGHT7HATISTHELENGTHOFTHE
SHORTERBASE
& METERS
( METERS
3TORE
!NSWEREACHOFTHEFOLLOWING
,OOKATTHEAPPLICATIONINTHECONCEPTMAP7HYMUSTTHEEXPRESSION
BEFACTOREDBEFOREUSINGTHE:ERO0RODUCT0ROPERTY
!NEWSTOREISBEINGBUILTINTHESHAPEOFARECTANGLEWITHAPARKING
LOTINTHESHAPEOFANISOSCELESTRAPEZOID4HEPARKINGLOTANDTHE
STOREWILLSHAREASIDEASSHOWN3ELECTTHEBESTANSWER
0ARKINGLOT
XORX
XX
£ÊÃiVœ˜`
XX
XX
)FSTDSTDTHENTORT
$URINGAGAMEOFGOLF+AYLEYHITSHER
BALLOUTOFASANDTRAP4HEEQUATION
HTTMODELSTHEHEIGHT
OFTHEGOLFBALLINFEETINRELATIONTOTHE
NUMBEROFTSECONDSSINCEITWASHIT
&INDHOWLONGITTAKES+AYLEYSGOLFBALL
TOREACHTOGREEN
£°xÊÃiVœ˜`Ã
!PPLICATION
5SETHE:ERO0RODUCT0ROPERTY
TOSOLVEQUADRATICEQUATIONS
)FXSXDTHENXORX
ÓääÊvÌ]Ê£ÓäÊvÌ]Ê£ÈäÊvÌ
2OBERTTHREWAROCKOFFABRIDGEINTOTHE
RIVER4HEDISTANCEFROMTHEROCKTOTHE
RIVERISMODELEDBYTHEEQUATION
HTTWHEREHISTHE
HEIGHTINFEETANDTISTHETIMEIN
SECONDS&INDHOWLONGITTOOKTHEROCK
TOENTERTHEWATER
AK4up.indd 86
œÌʏ}iLÀ>Ê£
7ORD$EFINITION
)FYOUMULTIPLYTWOORMORE
QUANTITIESANDTHEANSWERIS
ATLEASTONEOFTHEQUANTITIES
MUSTBE
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
Ó]ÊÓ]ÊÊÚÚÚ
Ê££ÊÊÊ
Ó
,i>`ˆ˜}Ê-ÌÀ>Ìi}ˆiÃ
5SEA#ONCEPT-AP
5SETHISCONCEPTMAPTOUNDERSTANDTHE:ERO0RODUCT0ROPERTY
!NARCHITECTISDESIGNINGABUILDINGWITH
ARIGHTTRIANGULARFOOTPRINT
(EIGHTOF!CORN
X XX
Î]ÊÎ]ÊÇ
xÊ]ÊÊÚÚ
ä]ÊÊÊÚÚ
Ê£Ê
Ó
Î
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
X X X
4HEHEIGHTOFANACORNFALLINGOUTOF
ATREEISHTWHEREHIS
HEIGHTINFEETANDTISTIMEINSECONDS
$ETERMINEHOWLONGITTAKESTHEACORN
TOREACHTHEGROUND#HECKYOUR
ANSWERBYGRAPHINGTHEFUNCTION
Ó
X X
7RITETHECORRECTANSWER
Ó Ó
*ÀœLi“Ê-œÛˆ˜}
3OLVING1UADRATIC%QUATIONSBY&ACTORING
ÎÊ]ÊÊÚÚ
ÚÚ
ÊxÊ]ÊÊÊÚÚ
ÊÎÊ
X X X
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
Î]Êx]Êx
X X X
X X
X X
X X X
* œÌʏ}iLÀ>Ê£
XX
SXDSXD
XÊÊx]ÊXÊÊÓ
#OPYRIGHT©BY(OLT2INEHARTAND7INSTON
!LLRIGHTSRESERVED
86
XÊÊ{]ÊXÊÊÎ
XX
XÊÊÓ]ÊXÊÊx
œÌʏ}iLÀ>Ê£
Holt Algebra 1
12/29/05 2:16:01 PM