1.09 Radical Equations
College
Algebra: One Step at a Time.
Pages
138 - 147: #11, 13, 15, 16, 19, 20, 21, 22, 23, 27, 29, 31, 33, 35,
37.
Dr. Robert J. Rapalje
Seminole State College of Florida
Altamonte Springs Campus
To see
Section 1.09, with
detailed explanations, examples, exercises, and answers,
click here!
11.
The first step in solving radical equations is to
isolate the radical. In this case, the radical is alone on the left side of
the equation, so it is already isolated. You may proceed by squaring both
sides, in order to “undo” the radical.
On the left side, when you square the square root, you
simply remove the radical sign. On the right side, remember that you are
squaring a binomial, which is the first squared, twice the product, and the
last squared:
At first, it looks like a quadratic equation, but
subtract 
from each side, and the term
subtracts out leaving
To solve for x, add
+4x
to
each side, and subtract 7 from each side, to get the x terms
on the left side, and the number terms on the right side:
x = -3
Since you squared both sides, the answer is NOT
guaranteed. You must check the answers:
Check: 
x = -3
It checks!
13.
Since the radical term is isolated on the right side of
the equation, you may proceed by squaring both sides, in order to “undo” the
radical.
On the left side, when you square the square root, you
simply remove the radical sign. On the right side, remember that you are
squaring a binomial, which is the first squared, twice the product, and the
last squared:
This one is a quadratic equation, so you must set it
equal to zero by subtracting 
and 
from each side:
Factor the right side of the equation, beginning by
factoring out the common factor of 2:
Since you squared both sides, the answers are NOT
guaranteed. You must check the answers:
Check:
It checks!
Check:
Reject this answer!!
The final answer is
15.
It’s a
good idea to begin by multiplying both sides of the equation by the
denominator which is 
in
order to clear the fractions.
Since the
radical term is isolated on the right side of the equation, you may proceed
by squaring both sides, in order to “undo” the radical.
On the
left side, when you square the square root, you simply remove the radical
sign. On the right side, remember that you are squaring a binomial, which
is the first squared, twice the product, and the last squared:
This is a
quadratic equation, so you must set it equal to zero by subtracting 
and
from
each side:
Factor the
right side of the equation by taking out the common factor of
:
Since you
squared both sides, the answers are NOT guaranteed. You must check the
answers:
Check:
It
checks!
Check:
It
also checks!!
Final
Answer: 
16.
It’s a
good idea to begin by multiplying both sides of the equation by the
denominator which is 
in
order to clear the fractions.
Since the
radical term is isolated on the right side of the equation, you may proceed
by squaring both sides, in order to “undo” the radical.
On the
left side, when you square the square root, you simply remove the radical
sign. On the right side, remember that you are squaring a binomial, which
is the first squared, twice the product, and the last squared:
This is a
quadratic equation, so you must set it equal to zero by subtracting
and
from
each side:
Factor the
right side of the equation:
Since you
squared both sides, the answers are NOT guaranteed. You must check the
answers:
Check:
It
checks!
Check:
It
also checks!!
Final
Answer: 
19.
You must first isolate one of the radicals on the left side. Try adding
.
Now, you can write this:
and square both sides
of the equation in order to eliminate the radical on the left side:
On the left side, you
just remove the square root sign. On the right side, square the first, take
twice the product, and square the second.
Isolate the radical on the right side by adding
and
to
each side:
Since there is a common
factor, divide both sides by 8.
Now, square both sides
again to eliminate the second radical.
This answer still must
be checked in the original equation:
This
checks!
The final answer is
20.
You can
square both sides of the equation in order to eliminate the radical on the
left side:
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
Isolate
the radical on the right side by adding
and
to
each side:
Since
there is a common factor, divide both sides by 2.
Now,
square both sides again to eliminate the second radical.
Set equal
to zero:
Solve the
quadratic equation, by factoring the common factor of x:
These
answers still must be checked in the original equation:
Check x =
0:
This
checks!!
Check x = 3:
This
also checks!!
The
final answer is
21.
The first
step is to isolate one of the radicals by adding
to each side:
Now, you
can square both sides of the equation in order to eliminate the radical on
the left side:
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
Isolate
the radical on the right side by adding
and 
to each side:
Since
there is a common factor, divide both sides by 4.
Now,
square both sides again to eliminate the second radical.
Set equal
to zero:
Solve the
quadratic equation, by factoring:
These
answers still must be checked in the original equation:
This checks!!
This one does NOT check.
The final
answer is 
22.
This is an
interesting problem! In my first attempts to solve this problem, I tried to
isolate the 
by adding 
to each side. This solution can be seen on page 305 of
my Intermediate Algebra book. However, I don’t know why, but it turns out
to be an easier solution if you isolate the
by subtracting
from each side.
You may
proceed by squaring both sides, in order to “undo” the radical on the left
side.
Of course
when you square the left side, you must square the negative, which is a
positive, and remove the radical sign. On the right side, remember that you
are squaring a binomial, which is the quantity times itself.
Subtract and 
from each side to isolate the other radical, and you have
Now,
square both sides of the racial.
Since you
squared both sides, the answers are NOT guaranteed. You must check the
answers:
Check:
x
= 0 
It
does NOT check!
x
= 8 
It
does NOT check!
Final Answer: NO SOLUTION!!
23.
The first
step is to isolate one of the radicals by subtracting from each side of the equation.
Next,
square both sides of the equation in order to eliminate the radical on the
left side:
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
Now, you
must isolate the radical on the right side by adding
and 
to each side
Now,
square both sides again to eliminate the second radical.
Set equal
to zero by subtracting 
from each side:
Of course
it factors! In this case, take out the common factor of
:
These
answers still must be checked.
Check:
It checks!!
Check:
It does NOT check!!
The final answer is
27.
The first
step is to square both sides of the equation in order to eliminate the
radical on the left side:
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
Isolate the radical on the right side by adding +6x and − 26 to each
side:
Since
there is a common factor, divide both sides by 2.
Now,
square both sides again to eliminate the second radical.
Set equal
to zero by adding 
to each side:
Solve the
quadratic equation! You can use factoring, the calculator program “POLYSMLT”,
or the quadratic formula! By factoring, it looks like this. It’s a
trinomial, so start by taking out the common factor of 2:
These
answers still must be checked, and it turns out that the
does NOT check. However, the
DOES check, since
.
.
29.
The first
step is to isolate one of the radicals by adding
to each side of the equation.
Next,
square both sides of the equation in order to eliminate the radical on the
left side:
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
The next
step is a small, but important one. Combine the
and the 
on the right side.
Now, you
must isolate the radical on the right side by adding
and to each side
Now,
square both sides again to eliminate the second radical.
Set equal
to zero by adding 
to each side:
Hopefully(??!!) it factors, but it can also be solved using the calculator
program “POLYSMLT” or the quadratic formula! The method of factoring is
given here.
These
answers still must be checked.
Check:
It checks!!
Check:
This also checks!
The final answer is
31.
The first step is to isolate one of the radicals by subtracting
from
each side of the equation.
Next, square both sides
of the equation in order to eliminate the radical on the left side:
On the left side, you
just remove the square root sign. On the right side, square the first, take
twice the product, and square the second.
The next step is a small, but important one. Combine the
and
the 
on
the right side.
Now, you must isolate the radical on the right side by subtracting
and
from
each side
Divide both sides of
the equation by 2:
Now, square both sides
again to eliminate the second radical.
Set equal to zero by adding to
each side:
Hopefully(??!!) it
factors, but it can also be solved using the calculator program “POLYSMLT”
or the quadratic formula! The method of factoring is given here.
These answers still
must be checked.
Check :
It
checks!!
Check 
:
This does NOT check!!
The final answer is .
33.
The first
step is to isolate one of the radicals by adding
to
each side of the equation.
Next,
square both sides of the equation in order to eliminate the radical on the
left side:
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
Now, you
must isolate the radical on the right side by adding
and
to
each side
You will
save yourself some work if you notice that you can divide each side by 3
before you square both sides to eliminate the second radical.
Now,
square both sides again to eliminate the second radical.
Set equal
to zero by adding 
to
each side:
Of course
it factors!!
This
answer still must be checked.
Check:
It checks!!
The final
answer is .
35.
Solution:
The
radical on the left side is already isolated, so the first step is to square
both sides of the equation in order to eliminate the radical on the left
side.
On the
left side, you just remove the square root sign. On the right side, square
the first, take twice the product, and square the second.
The next
step is a small, but important one. Combine the
and
the 
on
the right side.
Now, you
must isolate the radical on the right side by adding
and
to
each side
Now,
square both sides again to eliminate the second radical.
Set equal
to zero by adding 
to
each side:
Hopefully(??!!) it factors, but it can also be solved using the calculator
program “POLYSMLT” or the quadratic formula! The method of factoring is
given here.
These
answers still must be checked.
Check:
It DOES NOT check, so it is rejected!!
Check:
This
checks!
The final
answer is
37.
Solution:
First, isolate one of the radicals on the left side of the equation by
adding 
to
each side of the equation.
Then, square both sides
of the equation in order to eliminate the radical on the left side.
On the left side, you
just remove the square root sign. On the right side, square the first, take
twice the product, and square the second.
Eliminate the
parentheses using the distributive property.
The next step is a small, but important one. Combine the
and
the 
on
the right side.
Now, you must isolate the radical on the right side by adding
and
to
each side
Divide both sides by 2:
Now, square both sides
again to eliminate the second radical.
Set equal to zero by adding 
and
to
each side:
Hopefully(??!!) it
factors, but it can also be solved using the calculator program “POLYSMLT”
or the quadratic formula! The method of factoring is given here.
These answers still
must be checked.
Check:
It
DOES NOT check, so it is rejected!!
Check: 
(NOTE:
This is the hard one!! A calculator might help!!)
This
checks!!
The final answer is
A
very interesting (and simple!!)
solution can be obtained with a graphing calculator:
(Coming Soon!!)
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