Question

Solve for x and show your steps. Is the solution extraneous? Check your work to show how you determined if the solution is extraneous or not.

Answer 1

Given the following equation:

`\text{ }\sqrt{3x\text{ + 12}}\text{ = 9}`

Let's solve for x,

`\text{ }\sqrt{3x\text{ + 12}}\text{ = 9}`
`\text{ \lparen}\sqrt{3x\text{ + 12}})^2\text{ = \lparen9\rparen}^2\text{ }`
`\text{ 3x + 12 = 81}`
`\text{ 3x = 81 - 12}`
`\text{ 3x = 69}`
`\text{ }\frac{\text{ 3x }}{\text{ 3 }}\text{ = }\frac{\text{ 69 }}{\text{ 3 }}`
`\text{ x = 23}`

Therefore, x = 23

In mathematics, an extraneous solution (or spurious solution) is a solution, such as that to an equation, that emerges from the process of solving the problem but is not a valid solution to the problem.

To check if the solution is extraneous, plug in x = 23 in the equation.

`\begin{gathered} \text{ }\sqrt{3x\text{ + 12}}\text{ = 9} \\ \text{ }\sqrt{3(23)\text{ + 12}}\text{ = 9} \\ \text{ }\sqrt{69\text{ + 12}}\text{ = 9} \\ \text{ }√(81)\text{ = 9} \\ \text{ 9 = 9 \lparen True\rparen} \end{gathered}`

The solution is, therefore, not extraneous.