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Extraneous solutions of radical equations

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.
x=x+2+7
His first few steps are given below.
x7=x+2(x7)2=(x+2)2x214x+49=x+2
Is it necessary for Caleb to check his answers for extraneous solutions?
Choose 1 answer:

Practice question 2

Elena is solving the following equation for x.
A3x53+2=7
Her first few steps are given below.
A3x53=5(A3x53)3=(5)33x5=125
Is it necessary for Elena to check her answers for extraneous solutions?
Choose 1 answer:

Practice question 3

Addison solves the equation below by first squaring both sides of the equation.
2x1=8x
What extraneous solution does Addison obtain?
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Practice question 4

Which value for the constant d makes x=1 an extraneous solution in the following equation?
8x=2x+d
d=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want to join the conversation?

  • blobby green style avatar for user Isabella
    is 0 an extraneous solution
    (18 votes)
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  • leaf green style avatar for user Abd Hamid Mat Sain
    square root 9 should be either 3 or -3.......why should -3 be extraneous then?
    (8 votes)
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  • leaf red style avatar for user Carla Cristina Almeida
    In the last situation, why did they plug -1 in the equation? was it a random number?
    (10 votes)
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  • spunky sam red style avatar for user Robert Roller
    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?
    (7 votes)
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    • aqualine ultimate style avatar for user Himalaya
      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)
  • blobby green style avatar for user 1earth4ever
    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.
    (6 votes)
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    • primosaur seed style avatar for user Ian Pulizzotto
      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!
      (8 votes)
  • winston default style avatar for user Tetsuya
    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)
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    • leaf green style avatar for user kubleeka
      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.
      (9 votes)
  • leaf green style avatar for user Sally
    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??
    (4 votes)
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  • aqualine sapling style avatar for user Joslynkim75
    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.
    (3 votes)
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    • stelly blue style avatar for user Kim Seidel
      This is a known error. A correction box does pop up on the video at about and 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.
      (4 votes)
  • primosaur seedling style avatar for user Courtney Ellwein
    In question 4 why can't the square root of 9 = -3 ?
    (2 votes)
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  • piceratops sapling style avatar for user Eliza
    I still don't get the reasoning behind the seemingly arbitrary decision to make the radical only refer to the principle root?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      It's not arbitrary.
      Apparently, you missed the key difference between the principle root and the negative root when first learning square root. Every math operation creates 1 result. A convention was established hundreds of years ago that if there is no sign or a + sign in front of the square root, then the problem is asking your for the principle root.
      sqrt(9) = +sqrt(9) = 3
      If a problem has a minus in front of the square root, then it is asking you for the negative root.
      -sqrt(9) = -3

      If this rule didn't exist, then if you were given the problem
      5+sqrt(9), there wouldn't be one answer. But, mathematicians want to keep things simple - each math operations creates one result.

      The one exception I know to this rule is when you apply a square root to solve a quadratice (this reverses the process of squaring something. Since squareing a number can result in the sign changing, we use both the principle and negative roots to make sure we have found all possible solutions.

      In these radical equations, the square root is already there. So, the problem tells you specifically which root it wants you to use.

      Hope this helps.
      (4 votes)