Pre-Calculus 6.0
SOLVING EQUATIONS CONTAINING RADICALS
KEY STEPS FOR EQUATIONS WITH ONE RADICAL:
1) Isolate the radical
2) “Liberate” the radicand
i.e. undo a square root by squaring, undo a cube root by cubing, etc.
3) Solve for the variable
4) Check your answer(s)
“BEWARE OF THE SQUARE” – squaring an expression (or raising to
an even power) may introduce EXTRANEOUS ROOTS (algebraic
solutions that do not work in the original problem).
Example 1:
Example 2:
1
4x + 8 + 9 =
10
2
2x + 7 − x =
2
1) Isolate the radical:
2x + 7 − x =
2
+x +x
1) Isolate the radical:
1
4x + 8 + 9 =
10
2
−9 −9
(2)
1
4x + 8 =
1 (2)
2
4x + 8 =
2
2) “Liberate” the radicand:
(
2x + 7 = x + 2
4x + 8
2) “Liberate” the radicand:
(
)
2
=( x + 2 )
2
2x + 7 = x2 + 4x + 4
3) Solve: Quadratic - try factoring and Zero Product Property
)
2
2x + 7 = x2 + 4x + 4
−7
−7
=
( 2)
2
4x + 8 =
4
3) Solve:
2x + 7
2x = x2 + 4x − 3
− 2x
− 2x
4x + 8 =
4
−8 −8
4x = − 4
x = −1
4) Check answer for extraneous roots:
?
1
4(−1) + 8 + 9 =
10
2
?
1
4 + 9=
10
2
?
1
10
( 2) + 9 =
2
1+ 9 =
10
TRUE so x =
{−1}
0 = x2 + 2x − 3
0 =( x + 3)( x − 1)
+3 0
−1 0
x=
x=
−3
x=
x=
1
Factor
Zero
Product
Property
4) Check answers for extraneous roots:
?
2(−3) + 7 − (−3) =2
?
1 − (−3) =2
?
1+ 3=
2
FALSE so − 3 is an extraneous root
Similar check of 1 yields TRUE statement
so, x = {1}
Pre-Calculus 6.0
SOLVING EQUATIONS CONTAINING RADICALS
KEY STEPS FOR EQUATIONS WITH TWO RADICALS:
1)
2)
3)
4)
5)
6)
Isolate one of the radicals (your choice)
“Liberate” the radicand of the isolated radical
Isolate the second radical
“Liberate” the radicand of the second radical
Solve for the variable
Check your answer(s)
Example 3:
2x + 6 − x + 4 =
1
1) Isolate one radical:
2 x + 6 =1 + x + 4
2) “Liberate” the radicand of the isolated radical:
(
2x + 6
) =(1 +
2
x+4
)
2
2 x + 6 =1 + 2 x + 4 + ( x + 4)
3) Isolate second radical:
x +=
1 2 x+4
4) “Liberate” the second radicand:
( x + 1) =
2
(2
x+4
)
2
x 2 + 2 x + 1= 4( x + 4)
5) Solve:
x 2 + 2 x + 1= 4( x + 4)
x 2 − 2 x − 15 =
0
( x − 5)( x + 3) =
0
x = − 3, 5
6) Check answer for extraneous roots:
-3 does not check - extraneous root
5 does check in original equation
x = {5}