3OLVE 2ADICAL %QUATIONS
'OAL + 3OLVE RADICAL EQUATIONS
9OUR .OTES
6/#!"5,!29
2ADICAL EQUATION ˜ÊiµÕ>̈œ˜Ê܈̅Ê>ÊÀ>`ˆV>Ê̅>ÌÊ
…>ÃÊÛ>Àˆ>LiÃʈ˜Ê̅iÊÀ>`ˆV>˜`
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4O SOLVE A RADICAL EQUATION FOLLOW THESE STEPS
3TEP Ê Ãœ>ÌiÊ THE RADICAL ON ONE SIDE OF THE
EQUATION IF NECESSARY
3TEP 2AISE EACH SIDE OF THE EQUATION TO THE SAME
Ê «œÜiÀÊ TO ELIMINATE THE RADICAL AND OBTAIN A
LINEAR QUADRATIC OR OTHER POLYNOMIAL EQUATION
3TEP Ê -œÛiÊ THE POLYNOMIAL EQUATION USING
TECHNIQUES YOU LEARNED IN PREVIOUS CHAPTERS
#HECK YOUR SOLUTION
%XAMPLE 3OLVE A RADICAL EQUATION
]
z
]
z
]
Ó
z
3OLVE q X qX ­ÊqÝÊÊÈÊ®Ê
Ê ÎÓÊ
Ê ÝÊÊÈÊ Ê ™Ê
Ê ÝÊ Ê ÎÊ
7RITE ORIGINAL EQUATION
3QUARE EACH SIDE TO ELIMINATE THE
RADICAL
3IMPLIFY
3UBTRACT Ê ÈÊ FROM EACH SIDE
4HE SOLUTION ISÊÊ ÎÊ #HECK THIS IN THE ORIGINAL EQUATION
#HECKPOINT 3OLVE THE EQUATION #HECK YOUR SOLUTION
]
q X z z
Î
,ESSON s !LGEBRA .OTETAKING 'UIDE
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9OUR .OTES
3OLVE AN EQUATION WITH A RATIONAL EXPONENT
%XAMPLE X /RIGINAL EQUATION
2AISE EACH SIDE TO THE
Ê Q­ÎÝÊÊ{®ÓÉÎRÎÉÓÊ Ê £ÈÎÉÓÊ
POWER ] z
Ê ÎÝÊÊ{Ê Ê ­£È£ÉÓ®ÎÊ !PPLY PROPERTIES OF
EXPONENTS
Ê ÎÝÊÊ{Ê Ê Ê È{Ê
3IMPLIFY
Ê ÎÝÊ ÊÊÊ ÈäÊ Ê
3UBTRACT Ê {Ê FROM EACH SIDE
Ê ÝÊ ÊzÊ ÓäÊ Ê
$IVIDE EACH SIDE BY Ê ÎÊ 4HE SOLUTION IS Ê ÓäÊ #HECK THIS IN THE ORIGINAL EQUATION
3OLVE AN EQUATION WITH AN EXTRANEOUS SOLUTION
%XAMPLE ]
X q X z
]
®ÊÓÊ
Ê ­ÝÊÊÓ®ÓÊ Ê ­ÊqÝÊÊ£äÊz
/RIGINAL EQUATION
3QUARE EACH SIDE
Ê Ý ÓÊÊ{ÝÊÊ{Ê Ê ÝÊÊ£äÊ
%XPAND LEFT SIDE AND
SIMPLIFY RIGHT SIDE
Ê Ý ÓÊÊxÝÊÊÈÊ 7RITE IN STANDARD
FORM
Ê ­ÝÊÊÈ®­ÝÊÊ£®Ê Ê &ACTOR
Ê ÝÊzÈÊ :ERO PRODUCT PROPERTY
OR Ê ÝÊz£Ê X zÊ ÈÊ OR
#(%#+
#HECK X Ê ÈÊ ]
X q X z
]
Ê
Ê ÈÊÊÓÊ gg q ÈÊÊ£äÊz
]
Ê {Ê gg q £ÈÊzÊ
Ê {Ê zzÊ {Ê
X Ê £Ê
3OLVE FOR X
#HECK X Ê £Ê ]
X q X z
]
Ê
Ê £ÊÊÓÊ ggg q £ÊÊ£äÊz
]
Ê ÎÊ ggg q™ÊzÊ
Ê ÎzpzÎÊ
4HE ONLY SOLUTION IS Ê {Ê 4HE APPARENT SOLUTION Ê £Ê IS
EXTRANEOUS
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,ESSON s !LGEBRA .OTETAKING 'UIDE
9OUR .OTES
3OLVE AN EQUATION WITH TWO RADICALS
%XAMPLE ]
z
]
z
]
3OLVE q X q Xz
]
qX q Xz
]
]
®ÊÓ
Ê ­ÊqÝÊÊÈÊz
ÊÊÓ®ÓÊ ­Êq £äÊÊÎÝÊz
]
ÝÊÊÈÊÊ{ÊqÝÊÊÈÊz
ÊÊ{Ê Ê £äÊÊÎÝÊ
]
Ê Ê {ÝÊ
Ê {ÊqÝÊÊÈÊz
]
Ê Êq ÝÊÊÈÊz
Ê ÊÊÊ ÝÊ
Ê
]
®ÊÓÊ Ê ­Ý®ÓÊ
Ê ­Êq ÝÊÊÈÊz
Ê ÝÊÊÈÊ Ê Ý ÓÊ
Ê Ý ÓÊÊÝÊÊÈÊ
7RITE ORIGINAL
EQUATION
3QUARE EACH
SIDE
%XPAND LEFT
SIDE AND
SIMPLIFY RIGHT
SIDE
)SOLATE RADICAL
EXPRESSION
$IVIDE EACH
SIDE BY 3QUARE EACH
SIDE AGAIN
3IMPLIFY
7RITE IN
STANDARD FORM
Ê ­ÝÊÊή­ÝÊÊÓ®Ê &ACTOR
Ê ÝÊzÎÊ Ê X Ê ÎÊ OR
#(%#+
Ê
:ERO PRODUCT
PROPERTY
OR Ê ÝÊzÓÊ X Ê ÓÊ ÊÊ
3OLVE FOR X
#HECK X Ê ÎÊ ]
]
Êq
Ê ÎÊÊÈÊz
ÊÊÓÊ ÊÊg q £äÊÊέήÊz
Ê
]
]
Ê Êq ™ÊÊz ÊÓÊ gg q £ÊzÊ
Ê xÊ zzÊ £Ê
#HECK X Ê ÓÊ ]
]]
ÊÊÓÊ Êgg q£äÊÊέӮÊz
Ê
Êq
Ê ­Ó®ÊÈÊz
]
]
Êq
Ê {ÊzÊÊÓÊ ggg q £ÈÊzÊ
Ê {Ê zzÊ {Ê
4HE ONLY SOLUTION IS Ê ÓÊ 4HE APPARENT SOLUTION Ê ÎÊ
IS EXTRANEOUS
,ESSON s !LGEBRA .OTETAKING 'UIDE
#OPYRIGHT Ú -C$OUGAL ,ITTELL(OUGHTON -IFFLIN #OMPANY
9OUR .OTES
#HECKPOINT 3OLVE THE EQUATION #HECK FOR EXTRANEOUS
SOLUTIONS
X Ê
Ê n
]
X q X z
Ê
Ê £
]
]
q X z
q X z
Ê {
(OMEWORK
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,ESSON s !LGEBRA .OTETAKING 'UIDE