Solving Radical Equations

Solving Radical Equations

Have paper ready for notes.

Poll

How are we feeling today?

Multiple Choice

Here is a radical equation to solve:

 $\sqrt{x+1}=4$  
Which of the following values would make this a true statement when substituted in for x?

• 3

• 8

• 15

• 16

 $\sqrt{x+1}=4$  

Substitute  $x=15$  
 $\sqrt{15+1}=4$  
 $\sqrt{16}=4$ 
  $4=4$  is a true statement.
Therefore,  $x=15$  is a solution!

Multiple Choice

How about this one?

 $\sqrt{5x+1}=6$  
Which of the following values would make this a true statement when substituted in for x?

• 36

• 7

• 6

• 1

 $\sqrt{5x+1}=6$  

Substitute  $x=7$  
 $\sqrt{5\left(7\right)+1}=6$  
 $\sqrt{35+1}=6$  
 $\sqrt{36}=6$ 
  $6=6$  is a true statement.
Therefore,  $x=7$  is a solution!

We won't always be given possible solutions to try. And so, we will need some steps!

Steps for Solving Radical Equations

• Isolate the radical

• Raise both sides to the same power as the index. (Clears the radical.)

• Solve the equation

• Check your answer

Multiple Choice

Remember our first equation:

 $\sqrt{x+1}=4$  
Step One is to Isolate the square root. Is the radical isolated?

• Yes. The Square Root is already isolated.

• No. Subtract 1 from both sides.

• No. Square both sides.

• No. Take the square root of 4.

Multiple Choice

Step Two is to Square Both Sides which looks like this:

 $\left(\sqrt{x+1}\right)^2=\left(4\right)^2$  
Which of the following would result from this step?

•  $\sqrt{x+1}=16$  
  

•  $\sqrt{x^2+1}=16$  
  

•  $x+1=8$  
  

•  $x+1=16$  
  

Multiple Choice

So far, we have:

 $\sqrt{x+1}=4$  
 $\left(\sqrt{x+1}\right)^2=\left(4\right)^2$ 
 $x+1=16$   
What should we do next?

• Nothing. x=16 is the solution.

• Add 1 to both sides.

• Subtract 1 from both sides.

• Subtract 16 from both sides.

Completed Problem

 $\sqrt{x+1}=4$  
 $\left(\sqrt{x+1}\right)^2=\left(4\right)^2$  
 $x+1=16$  
 $x=15$  
Remember to check your answer!

Multiple Choice

Our second equation:

 $\sqrt{5x+1}=6$  
Is the radical isolated?

• Yes. The radical is isolated.

• No. Subtract 1 from both sides.

• No. Divide both sides by 5.

• No. Add 1 to both sides.

Multiple Choice

The radical is isolated.

 $\sqrt{5x+1}=6$  
What do I do next?

• Subtract 1 from both sides

• Raise both sides to the 2nd power.

• Divide both sides by 5.

• Add 1 to both sides.

Multiple Choice

So far we have:

 $\sqrt{5x+1}=6$  
 $\left(\sqrt{5x+1}\right)^2=\left(6\right)^2$  
What does my next step look like?

•  $5x+1=36$  
  

•  $5x+1=6$  
  

•  $25x+1=36$  
  

•  $25x+1=6$  
  

Multiple Choice

 $\sqrt{5x+1}=6$   

 $\left(\sqrt{5x+1}\right)^2=\left(6\right)^2$  
 $5x+1=36$  
What is the last thing to do?

• First subtract 1 from both sides and then divide by 5.

• First subtract 1 from both sides and then multiply by 5.
  

• First divide by 5 and then subtract 1 from both sides.
  

• First add 1 to both sides and then divide by 5.

Completed Problem

 $\sqrt{5x+1}=6$  
 $\left(\sqrt{5x+1}\right)^2=\left(6\right)^2$  
 $5x+1=36$  
 $5x=35$  
 $x=7$  

Remember to check your answer!

Multiple Choice

Given the equation:
 $\sqrt{3x-2}+2=6$  
What would you do first?

• Square both sides

• Divide both sides by 3

• Add 2 to both sides

• Subtract 2 from both sides

Multiple Choice

We isolate the radical by subtracting 2 from both sides:
 $\sqrt{3x-2}+2=6$  
to get  $\sqrt{3x-2}=4$  

Now what?

• Square both sides

• Divide both sides by 3

• Add 2 to both sides

• Subtract 2 from both sides

Multiple Choice

So far we have:
 $\sqrt{3x-2}+2=6$  
 $\sqrt{3x-2}=4$  

 $\left(\sqrt{3x-2}\right)^2=\left(4\right)^2$  
 $3x-2=16$  
Now what?

• Subtract 2 from both sides and then divide by 3

• Divide both sides by 3 then add 2 to both sides

• Add 2 to both sides and then divide by 3

• Multiply 3 to both sides and then add 2 to both sides

Completed Problem

 $\sqrt{3x-2}+2=6$  
 $\sqrt{3x-2}=4$  
 $\left(\sqrt{3x-2}\right)^2=\left(4\right)^2$  
 $3x-2=16$  
 $3x=18$  
 $x=6$  

Remember to check your answer!

Multiple Choice

Given the equation:

 $-3\sqrt{x+2}=18$  
What would the correct next step look like?

•  $\sqrt{x+2}=-6$  

•  $\sqrt{x+2}=21$  

•  $-3\sqrt{x}=16$  

•  $-3\sqrt{x}=20$  

Multiple Choice

So far we have:

 $-3\sqrt{x+2}=18$  
 $\sqrt{x+2}=-6$  
What would the correct next step look like?

•  $\sqrt{x}=-8$  

•  $\sqrt{x}=-4$  

•  $\left(\sqrt{x+2}\right)^2=\left(-6\right)^2$  

•  $\left(\sqrt{x+2}\right)^3=\left(-6\right)^3$  

Multiple Choice

So far we have:

 $-3\sqrt{x+2}=18$  
 $\sqrt{x+2}=-6$  
 $\left(\sqrt{x+2}\right)^2=\left(-6\right)^2$  
 $x+2=36$  
Solve and check your solution. What would the correct answer be?

• No solution. The check failed.

•  $x=34$  

•  $x=38$  

•  $x=18$  

Remember to check your Solution!

Original Problem:  $-3\sqrt{x+2}=18$  

When we solved, we got x = 34.

 $-3\sqrt{\left(34\right)+2}=18$  
 $-3\sqrt{36}=18$  
 $-3\left(6\right)=18$  
 $-18=18$  which is not not a true statement.
Therefore, there is no solution to this problem.

Multiple Choice

What do we call a solution that does not work when you plug it back in to check?

• Extraordinary Solution

• Extraneous Solution

• Involuntary Solution

• Unnecessary Solution

We could have stopped earlier and said no solution:

Original Problem:  $-3\sqrt{x+2}=18$  

When we divided by -3, we got:

 $\sqrt{x+2}=-18$  
It is impossible to take the square root of a number and get a negative number as a result.

Get ready for the next problem.

Multiple Choice

What would you do first?

• Raise both sides to the 3rd power.

• Raise both sides to the 2nd power.

• Add 1 to both sides.

• Divide both sides by 3.

Multiple Choice

We raise both sides to the 3rd power. What would it look like afterwards?

•  $3x-1=4$  
  

•  $3x-1=2$  
  

•  $3x-1=8$  
  

•  $27x-1=8$  
  

Multiple Choice

Solve and check.

What would the answer be?

•  $x=3$  
  

•  $x=\frac{7}{3}$  
  

•  $x=-3$  
  

•  $x=\frac{8}{3}$  
  

The Whole Solution:

And our check works!

Poll

On your own, try to solve this problem using the steps.

Which of the following best describes your progress?

• I was able to isolate the radical, but then got stuck.

• I do not know where to start.

• I think I got a solution using the steps.

• I found a solution, but not using the steps.

• I got stuck somewhere in the middle.

I shall work through the problem on the shared screen.

Poll

On your own, try to solve this problem using the steps.

Which of the following best describes your progress?

• I was able to isolate the radical, but then got stuck.

• I do not know where to start.

• I think I got a solution using the steps.

• I found a solution, but not using the steps.

• I got stuck somewhere in the middle.

I shall work through the problem on the shared screen.