we are given the following expression:
![a\sqrt[]{x+b}+c=d](https://tex.z-dn.net/?f=a%5Csqrt%5B%5D%7Bx%2Bb%7D%2Bc%3Dd)
We are asked to find constants a, b, c, and d such that we get an extraneous solution and a non-extraneous solution.
Let's remember that an extraneous solution arises when solving a problem we reduce it to a simpler problem and get a solution but when replacing that solution in reality it's not a solution to the problem because it is undetermined or outside the domain of the original problem.
Part 1. Let's choose the following values:
![\begin{gathered} d=4 \\ c=8 \\ a=2 \\ b=-5 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20d%3D4%20%5C%5C%20c%3D8%20%5C%5C%20a%3D2%20%5C%5C%20b%3D-5%20%5Cend%7Bgathered%7D)
We get the equation:
![2\sqrt[]{x-5}+8=4](https://tex.z-dn.net/?f=2%5Csqrt%5B%5D%7Bx-5%7D%2B8%3D4)
Now let's take the following values for the constants:
![\begin{gathered} c=4 \\ d=8 \\ a=2 \\ b=-5 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20c%3D4%20%5C%5C%20d%3D8%20%5C%5C%20a%3D2%20%5C%5C%20b%3D-5%20%5Cend%7Bgathered%7D)
We get the equation:
![2\sqrt[]{x-5}+4=8](https://tex.z-dn.net/?f=2%5Csqrt%5B%5D%7Bx-5%7D%2B4%3D8)
Part 2. To get the extraneous solution we will isolate the radical first from the expression. To do that we will subtract "8" from both sides:
![2\sqrt[]{x-5}=4-8](https://tex.z-dn.net/?f=2%5Csqrt%5B%5D%7Bx-5%7D%3D4-8)
Now we'll divide by "2":
![\sqrt[]{x-5}=\frac{4-8}{2}](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7Bx-5%7D%3D%5Cfrac%7B4-8%7D%7B2%7D)
Let's choose the following values:
![\begin{gathered} d=4 \\ c=8 \\ a=2 \\ b=-5 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20d%3D4%20%5C%5C%20c%3D8%20%5C%5C%20a%3D2%20%5C%5C%20b%3D-5%20%5Cend%7Bgathered%7D)
Now let's solve for "x":
![\sqrt[]{x-5}=-\frac{4}{2}](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7Bx-5%7D%3D-%5Cfrac%7B4%7D%7B2%7D)
![\sqrt[]{x-5}=-2](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7Bx-5%7D%3D-2)
Elevating both sides to the second power:
![(\sqrt[]{x-5})^2=(-2)^2](https://tex.z-dn.net/?f=%28%5Csqrt%5B%5D%7Bx-5%7D%29%5E2%3D%28-2%29%5E2)
Solving:
![x-5=4](https://tex.z-dn.net/?f=x-5%3D4)
Adding 5 on both sides:
![\begin{gathered} x=4+5 \\ x=9 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D4%2B5%20%5C%5C%20x%3D9%20%5Cend%7Bgathered%7D)
Now that we get a solution we need to check it by replacing the value we found for "x" in the initial equation:
![\sqrt[]{x-5}=-2](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7Bx-5%7D%3D-2)
Replacing the value of "x":
![\sqrt[]{9-5}=-2](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7B9-5%7D%3D-2)
Solving the operation inside the radical:
![\sqrt[]{4}=-2](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7B4%7D%3D-2)
Solving the radical:
![2=-2](https://tex.z-dn.net/?f=2%3D-2)
Now we use the second equation:
![2\sqrt[]{x-5}+4=8](https://tex.z-dn.net/?f=2%5Csqrt%5B%5D%7Bx-5%7D%2B4%3D8)
Isolating the radical we get
![\sqrt[]{x-5}=\frac{8-4}{2}](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7Bx-5%7D%3D%5Cfrac%7B8-4%7D%7B2%7D)
Solving the operations:
![\begin{gathered} \sqrt[]{x-5}=\frac{4}{2} \\ \sqrt[]{x-5}=2 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Csqrt%5B%5D%7Bx-5%7D%3D%5Cfrac%7B4%7D%7B2%7D%20%5C%5C%20%5Csqrt%5B%5D%7Bx-5%7D%3D2%20%5Cend%7Bgathered%7D)
Squaring both sides:
![(\sqrt[]{x-5})^2=(2)^2](https://tex.z-dn.net/?f=%28%5Csqrt%5B%5D%7Bx-5%7D%29%5E2%3D%282%29%5E2)
Solving the square:
![x-5=4](https://tex.z-dn.net/?f=x-5%3D4)
adding 5 on both sides:
![\begin{gathered} x=4+5 \\ x=9 \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20x%3D4%2B5%20%5C%5C%20x%3D9%20%5Cend%7Bgathered%7D)
Now, replacing the value of "x" in the original equation:
![\sqrt[]{9-5}=2](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7B9-5%7D%3D2)
Solving the operation inside the radical:
![\sqrt[]{4}=2](https://tex.z-dn.net/?f=%5Csqrt%5B%5D%7B4%7D%3D2)
Solving the radical.
![2=2](https://tex.z-dn.net/?f=2%3D2)
Therefore, x = 9 is a solution to this equation.
Part 3. Since the value we found for "x" in the first equation does not give a solution, this means that x = 9 is an extraneous solution for the first given values of the constants.