3.6 Solve Radical Equations
p.204
How do you solve a radical equation?
What must you do when you solve an
equation with two radicals?
What is an extraneous solution?
Key Concept
**
The nth power function and the nth root
function are inverses. So raising an nth
root to the nth power gets rid of the
radical symbol. Also taking the nth root
of something raised to the nth power
gets rid of the power
Ex: Solve for x.
Don’t forget to
check your
solutions!!
**
Use this process to solve the radical equations.
Write original equation.
Cube each side to eliminate the radical.
2x+7 = 27
2x = 20
x = 10
Simplify.
Subtract 7 from each side.
Divide each side by 2.
Check x = 10 in the original equation.
Substitute 10 for x.
Solution checks out.
We are done!
Ex: Solve for y.
Don’t forget to
check your
solutions!!
Ex: Solve.
Why plus or minus?
We are taking an
even nth root with a
variable
Don’t forget to
check your
solutions!!
Ex: Solve.
Don’t forget to
check your
solutions!!
Ex: Solve.
Factor this to solve!
Solution
Don’t forget to
check your
solutions!!
Only use
the positive
answers
when
checking
solutions.
So, why didn’t one of the
solutions work?
• The solution that did not work is called
an extraneous solution.
• An extraneous solution is a false
solution.
• It can happen when you raise each side
of an equation to the same power.
• This is the reason why you MUST
check all solutions!!
Solve x + 1 = √7x + 15 .
x + 1 = √ 7x + 15
(x + 1)2 = (√7x + 15)2
x2 + 2x + 1 = 7x + 15
x2 – 5x – 14 = 0
(x – 7)(x + 2) = 0
x – 7 = 0 or x + 2 = 0
x = 7 or
x = –2
Write original equation.
Square each side.
Expand left side and simplify
right side.
Write in standard form.
Factor.
Zero-product property
Solve for x.
CHECK 1
Check x = 7 in the
original equation.
Check x = –2 in the
original equation.
x + 1 = √7x + 15
x + 1 = √7x + 15
7 + 1 =? √ 7(7) + 15
–2 + 1 =? √ 7(–2) + 15
8 =? √ 64
–1 =? √ 1
8=8
–1 =/ 1
ANSWER
The only solution is 7.
(The apparent solution −2 is extraneous.)
Solve the equation. Check your solution.
6.
–2x3/4 = –16
–2x3/4 = –16
Write original equation.
x3/4 = 8
Divide each side by –2.
x = (81/3)4
Raise each side to the power 4 .
3
Apply properties of exponent.
x = 16
Simplify.
(x3/4)4/3 = 84/3
Solve the equation. Check your solution.
8.
(x + 3)5/2 = 32
(x + 3)5/2 = 32
[(x+ 3)5/2]2/5 = 322/5
Write original equation.
Raise each side to the power 2/5.
Apply properties of exponent.
x+3=4
Simplify.
x=1
Simplify.
• How do you solve a radical equation?
Isolate the radical on one side of the equation and
raise the equation to the power in the problem
to eliminate the radical.
• What must you do when you solve an equation
with two radicals?
Put a radical on each side of the equal sign before
raising to the power of the radical.
• What is an extraneous solution?
A answer that is not a valid solution of the original
problem.
3.6 Assignment
Page 208 #5, 6-12 even,
13, 14-18 even, 25-28,
35-37, 45-51 odd
BONUS: 53, 60