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

Sanford, FL  32773

 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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