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## Get ready for AP® Calculus

# Extraneous solutions of radical equations

CCSS.Math:

Practice some problems before going into the exercise.

## Introduction

In this article we will practice some problems that involve thinking about the conditions for obtaining extraneous solutions while solving radical equations.

### Practice question 1

Caleb is solving the following equation for x.

His first few steps are given below.

### Practice question 2

Elena is solving the following equation for x.

Her first few steps are given below.

### Practice question 3

Addison solves the equation below by first squaring both sides of the equation.

### Practice question 4

## Want to join the conversation?

- is 0 an extraneous solution(18 votes)
- It doesn't have to be. For example, x^2+4x=0 has solutions 0 and -4 (try them)(18 votes)

- So take the following as an example.

sqrt(4x + 41) = x+5

Originally, x must be greater than -41/4 to be valid. When we solve this equation algebraically, we get the following two solutions: X = -8 and X = 2.

When you plug X = -8 back into the equation, you end up with sqrt(9) = -3.

This is extraneous because we are supposed to use the principle root.

I have three questions.

#1: Why do we have to use the principle root? It seems logical that the sqrt(9) should be equal to both +3 and -3.

#2: Why is this extraneous solution still within the original domain of the function, as it is > -41/4.

#3: What happened mathematically to produce this extraneous solution. If you graph the two equations separately, they only intersect at x = 2, therefore x = 8 is not a solution. But why is this? What happened to produce this extra answer?(8 votes)- 1. In your example, we use the principal square root because the original problem statement says so. It says √(4x + 41), and not -√(4x + 41). By definition, the radical notation without a minus or ± sign in front of it means “the principal root”, which is always positive. The solutions to Power Equations and the solutions to Radical Equations are different things. For example, x^2 = 4 has two solutions: x = ±2, while x = √4 has only one solution: x = 2, and x = -√4 also has one solution: x = -2. We define the square root as a function, so it must have only one output for each input. If we define the function y = √x as having two solutions, then it is no longer a function.

2. The solution x = -8 is extraneous to the original equation √(4x + 41) = x + 5. However, it is the solution to the equation -√(4x + 41) = x + 5. The expression under the radical is same in both equations, so in terms of keeping the radicand non-negative, the value -8 is OK. If we take the function y = √(4x + 41), then -8 would be a valid input value for x. However, for the equation √(4x + 41) = x + 5, the value x = -8 is not a solution, because it leads to an invalid statement 3 = -3, which is not true.

3. As mentioned above, x = -8 is the solution to the equation -√(4x + 41) = x + 5. So, if we graph -√(4x + 41) instead of √(4x + 41), it will intersect with x + 5 at x = -8.(11 votes)

- In the last situation, why did they plug -1 in the equation? was it a random number?(8 votes)
- This got me at first too. In the original question, it specifically asks you to use x=-1(8 votes)

- I still don't understand why I should care about extraneous solution. It's outside the domain, not a solution, a wrong answer. Or is there any use of finding it?(5 votes)
- Extraneous solutions are not necessarily outside the domain. But they can appear as extra solutions when we square both sides of an equation, because when we square an equation, we would get the same result whether the original equation was positive or negative. So one solution corresponds to the positive equation, and the other to the negative equation. But both of them will fall out of the algebra. We need to be able to tell which solution is extraneous and which works.(8 votes)

- square root 9 should be either 3 or -3.......why should -3 be extraneous then?(7 votes)
- You are correct that square root (9) can be 3 or -3. But, there is a rule that we always use the principle root (the positive value) unless the square root has a minus in front of it.

sqrt(9) is asking for the principle root = 3

- sqrt(9) is asking for the negative root = -3.

See this video: https://www.khanacademy.org/math/algebra-basics/basic-alg-foundations/alg-basics-roots/v/introduction-to-square-roots(2 votes)

- I am having a really hard time with this unit. I have watched all the videos several times and I am still very confused. I would really appreciate some help.(5 votes)
- The key idea for solving square root equations is to isolate a radical on one side and square both sides. If there's still a radical in the equation, then this process would need to be performed a second time. (By the way, don't forget to include the middle term when squaring a binomial. Many students forget to do this.)

After you have solved the resulting linear or quadratic equation for x, remember that you're not finished yet! Because every positive number has a positive and a negative square root, but the radical symbol denotes only the positive (principal) square root, the act of squaring both sides can create invalid (extraneous) solutions! So plug in your solutions to the**original**equation to determine which solutions work and which solutions must be discarded.

Have a blessed, wonderful day!(5 votes)

- Hi I don't understand why √9 would not equal -3 when it equals 3. In practice question 3, why wouldn't x=-1 be a correct solution??(3 votes)
- I think they're looking for the principal sqrt of 9.(3 votes)

- In the video titled, "Equation that has a specific extraneous solution." The person who did the video squared the right side of the equation wrong. (d+2x)^2 should be d^2 + 4dx + 4x^2. This is an error.(2 votes)
- This is a known error. A correction box does pop up on the video at about1:05and tells you there is an error. However, these are only visible if you aren't watching in full screen mode. So, if you find something like this, you should always check to see if there is an error message.(3 votes)

- how do u check for the extraneous soluations ?(2 votes)
- Get all solutions you found and plug them in to the very original function(2 votes)

- For the question with Elena, doesn't she have to check too because it's possible to have no solutions?(2 votes)
- we squared the both sides by an odd number. So it's not necessary to check the equation(2 votes)