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Drag each equation to the correct location in the table. Determine which equations will result in extraneous solutions.

\begin{tabular}{|l|l|}
\hline
Extraneous Solutions & No Extraneous Solutions \\
\hline
[tex]$\sqrt{x} = 5$[/tex] & [tex]$\sqrt[3]{x} = 5$[/tex] \\
[tex]$\sqrt{x} = -5$[/tex] & [tex]$\sqrt[3]{x} = -5$[/tex] \\
[tex]$\sqrt[4]{x-2} = -2$[/tex] & [tex]$\sqrt[4]{x+3} = 4$[/tex] \\
[tex]$\sqrt[4]{x+1} = -2$[/tex] & \\
[tex]$\sqrt[7]{x+3} = -3$[/tex] & \\
\hline
\end{tabular}


Sagot :

To determine which equations result in extraneous solutions, let's analyze each given equation.

1. [tex]\(\sqrt{x} = 5\)[/tex]
2. [tex]\(\sqrt{x} = -5\)[/tex]
3. [tex]\(\sqrt[3]{x} = 5\)[/tex]
4. [tex]\(\sqrt[3]{x} = -5\)[/tex]
5. [tex]\(\sqrt[4]{x-2} = -2\)[/tex]
6. [tex]\(\sqrt[4]{x+3} = 4\)[/tex]
7. [tex]\(\sqrt[4]{x+1} = -2\)[/tex]
8. [tex]\(\sqrt[7]{x+3} = -3\)[/tex]

### Equations Resulting in Extraneous Solutions:
- [tex]\(\sqrt{x} = -5\)[/tex]: This is an extraneous solution because the square root of a number cannot be negative.
- [tex]\(\sqrt[4]{x-2} = -2\)[/tex]: This is an extraneous solution because the fourth root of a number cannot be negative.
- [tex]\(\sqrt[4]{x+1} = -2\)[/tex]: Again, the fourth root of a number cannot be negative.
- [tex]\(\sqrt[7]{x+3} = -3\)[/tex]: This is an extraneous solution because although the seventh root can be negative, solving this equation results in [tex]\(x\)[/tex] such that [tex]\(x+3\)[/tex] yields a real number whose seventh root is indeed [tex]\(-3\)[/tex].

Therefore, the equations that will result in extraneous solutions are:
[tex]\[ \sqrt{x} = -5, \quad \sqrt[4]{x-2} = -2, \quad \sqrt[4]{x+1} = -2, \quad \sqrt[7]{x+3} = -3 \][/tex]

### Equations Without Extraneous Solutions:
- [tex]\(\sqrt{x} = 5\)[/tex]: The square root of [tex]\(x\)[/tex] can be 5.
- [tex]\(\sqrt[3]{x} = 5\)[/tex]: The cube root of [tex]\(x\)[/tex] can be 5.
- [tex]\(\sqrt[3]{x} = -5\)[/tex]: The cube root of [tex]\(x\)[/tex] can be -5.
- [tex]\(\sqrt[4]{x+3} = 4\)[/tex]: The fourth root of [tex]\(x+3\)[/tex] can be 4.

Therefore, the equations without extraneous solutions are:
[tex]\[ \sqrt{x} = 5, \quad \sqrt[3]{x} = 5, \quad \sqrt[3]{x} = -5, \quad \sqrt[4]{x+3} = 4 \][/tex]

### Final Table:

[tex]\[ \begin{array}{|l|l|} \hline \text{Extraneous Solutions} & \text{No Extraneous Solutions} \\ \hline \sqrt{x} = -5 & \sqrt{x} = 5 \\ \sqrt[4]{x-2} = -2 & \sqrt[3]{x} = 5 \\ \sqrt[4]{x+1} = -2 & \sqrt[3]{x} = -5 \\ \sqrt[7]{x+3} = -3 & \sqrt[4]{x+3} = 4 \\ \hline \end{array} \][/tex]
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