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: 

32.       

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

33.       

 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     

34.       

 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

35.                       

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!!

36.       

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  

 40.                   

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

                     .

The final answer is .

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