Solve a Radical Equation in One Variable

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2.6 Radical Equations in One Variable

• ​ A radical equation is an equation that has at least one radical expression containing a variable, while any nonradical expressions are polynomial terms.

• In math, an extraneous solution is a solution that emerges during the process of solving a problem. It does not work when you plug it back in to check.

Multiple Choice

Examine the following radical expression. What is the index?

$$\sqrt[5]{108}$$  

• 5

• 108

• 36

Multiple Choice

Examine the following radical expression. What is the radicand?

$$\sqrt[5]{108}$$

• 5

• 108

• 36

Rational Exponents

Multiple Choice

Write the index/root: $$13\ ^{\frac{1}{4}}$$

• $$13$$

• $$4$$

• $$1$$

Multiple Choice

Write the exponent: $$13\ ^{\frac{1}{4}}$$

• $$13$$

• $$4$$

• $$1$$

Multiple Choice

Write the radicand: $$13\ ^{\frac{1}{4}}$$

• $$13$$

• $$4$$

• $$1$$

Multiple Choice

Write as a radical expression: $$13\ ^{\frac{1}{4}}$$

• $$\sqrt[13]{4}$$

• $$\sqrt[1]{13}$$

• $$\sqrt[4]{13}$$

Multiple Choice

Write the following in rational exponent (exponential) form:

$$\sqrt[3]{x^7}$$

• $$x^{\frac{7}{3}}$$

• $$3x^7$$

• $$x^{\frac{3}{7}}$$

• $$7x^3$$

Multiple Choice

Write the following in rational exponent (exponential) form:

$$\sqrt[6]{2}$$

• $$2^6$$

• $$2x^6$$

• $$2^{\frac{1}{6}}$$

• $$6x^2$$

Multiple Choice

Write the following in rational exponent (exponential) form:

$$\sqrt[]{7}$$

• $$7^2$$

• $$7x^2$$

• $$7^{\frac{1}{2}}$$

• $$2^7$$

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

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

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:

$$\left(\sqrt{x+1}\right)^2=\left(4\right)^2$$  
$$\sqrt{x+1}=4$$
$$x+1=16$$   
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

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

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:
$$\sqrt{3x-2}+2=6$$  
$$\sqrt{3x-2}=4$$  

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

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:

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

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

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:

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 board.

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 board.