Question

which of the following is an extraneous solution of \\((45-3x)^{\frac{1}{2}} = x-9\\)?

\\(x = -12\\)
\\(x = -3\\)
\\(x = 3\\)
\\(x = 12\\)

Explanation:

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<reasoning>

Square both sides of the equation

Using the Solving Radical Equations knowledge point
\[

$$\begin{aligned} (45-3x)^{\frac{1}{2}} &= x-9 \\ 45-3x &= (x-9)^2 \\ 45-3x &= x^2 - 18x + 81 \end{aligned}$$

\]

Solve the quadratic equation

Using the Polynomial Factoring knowledge point
\[

$$\begin{aligned} x^2 - 15x + 36 &= 0 \\ (x-12)(x-3) &= 0 \\ x = 12 \quad &\text{or} \quad x = 3 \end{aligned}$$

\]

Check for extraneous solutions

Using the Extraneous Solutions knowledge point
\[

$$\begin{aligned} &\text{For } x = 12:\\ &\text{LHS} = (45-3(12))^{\frac{1}{2}} = 9^{\frac{1}{2}} = 3\\ &\text{RHS} = 12 - 9 = 3 \quad (\text{Valid})\\ \\ &\text{For } x = 3:\\ &\text{LHS} = (45-3(3))^{\frac{1}{2}} = 36^{\frac{1}{2}} = 6\\ &\text{RHS} = 3 - 9 = -6 \quad (\text{Invalid, } 6 eq -6) \end{aligned}$$

\]
Thus, \(x = 3\) is the extraneous solution.
</reasoning>

<answer>
<mcq-option>x = -12</mcq-option>
<mcq-option>x = -3</mcq-option>
<mcq-correct>x = 3</mcq-correct>
<mcq-option>x = 12</mcq-option>
</answer>

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Answer:

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<reasoning>

Square both sides of the equation

Using the Solving Radical Equations knowledge point
\[

$$\begin{aligned} (45-3x)^{\frac{1}{2}} &= x-9 \\ 45-3x &= (x-9)^2 \\ 45-3x &= x^2 - 18x + 81 \end{aligned}$$

\]

Solve the quadratic equation

Using the Polynomial Factoring knowledge point
\[

$$\begin{aligned} x^2 - 15x + 36 &= 0 \\ (x-12)(x-3) &= 0 \\ x = 12 \quad &\text{or} \quad x = 3 \end{aligned}$$

\]

Check for extraneous solutions

Using the Extraneous Solutions knowledge point
\[

$$\begin{aligned} &\text{For } x = 12:\\ &\text{LHS} = (45-3(12))^{\frac{1}{2}} = 9^{\frac{1}{2}} = 3\\ &\text{RHS} = 12 - 9 = 3 \quad (\text{Valid})\\ \\ &\text{For } x = 3:\\ &\text{LHS} = (45-3(3))^{\frac{1}{2}} = 36^{\frac{1}{2}} = 6\\ &\text{RHS} = 3 - 9 = -6 \quad (\text{Invalid, } 6 eq -6) \end{aligned}$$

\]
Thus, \(x = 3\) is the extraneous solution.
</reasoning>

<answer>
<mcq-option>x = -12</mcq-option>
<mcq-option>x = -3</mcq-option>
<mcq-correct>x = 3</mcq-correct>
<mcq-option>x = 12</mcq-option>
</answer>

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