Question

study the steps shown to solve the given equation.
\\(\sqrt{30 - 2x} = x - 3\\)
\\(30 - 2x = x^2 - 6x + 9\\)
\\(0 = x^2 - 4x - 21\\)
\\(0 = (x + 3)(x - 7)\\)
based on the above work, possible solutions of the equation are dropdown.

Explanation:

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<reasoning>

Find the roots of the factored equation

Using the Zero Product Property knowledge point
\[

$$\begin{aligned} &0 = (x + 3)(x - 7) \\ &x + 3 = 0 \implies x = -3 \\ &x - 7 = 0 \implies x = 7 \end{aligned}$$

\]

Check for extraneous solutions

Using the Solving Radical Equations and Extraneous Solutions knowledge points
\[

$$\begin{aligned} &\text{For } x = -3:\\ &\text{LHS} = \sqrt{30 - 2(-3)} = \sqrt{36} = 6 \\ &\text{RHS} = -3 - 3 = -6 \\ &\text{LHS} eq \text{RHS} \implies x = -3 \text{ is extraneous.} \\ \\ &\text{For } x = 7:\\ &\text{LHS} = \sqrt{30 - 2(7)} = \sqrt{16} = 4 \\ &\text{RHS} = 7 - 3 = 4 \\ &\text{LHS} = \text{RHS} \implies x = 7 \text{ is a valid solution.} \end{aligned}$$

\]

Determine the possible solutions based on the steps

The algebraic steps shown lead to the factored equation \(0 = (x + 3)(x - 7)\). Based strictly on the algebraic work shown up to that point, the possible solutions (before checking for extraneous solutions) are \(x = -3\) and \(x = 7\).
</reasoning>

<answer>
Study the steps shown to solve the given equation.
\[\sqrt{30-2x} = x-3\]
\[30-2x = x^2-6x+9\]
\[0 = x^2-4x-21\]
\[0 = (x+3)(x-7)\]
Based on the above work, possible solutions of the equation are <blank>\(x = -3\) and \(x = 7\)</blank>.
</answer>

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</post_analysis>

Answer:

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<reasoning>

Find the roots of the factored equation

Using the Zero Product Property knowledge point
\[

$$\begin{aligned} &0 = (x + 3)(x - 7) \\ &x + 3 = 0 \implies x = -3 \\ &x - 7 = 0 \implies x = 7 \end{aligned}$$

\]

Check for extraneous solutions

Using the Solving Radical Equations and Extraneous Solutions knowledge points
\[

$$\begin{aligned} &\text{For } x = -3:\\ &\text{LHS} = \sqrt{30 - 2(-3)} = \sqrt{36} = 6 \\ &\text{RHS} = -3 - 3 = -6 \\ &\text{LHS} eq \text{RHS} \implies x = -3 \text{ is extraneous.} \\ \\ &\text{For } x = 7:\\ &\text{LHS} = \sqrt{30 - 2(7)} = \sqrt{16} = 4 \\ &\text{RHS} = 7 - 3 = 4 \\ &\text{LHS} = \text{RHS} \implies x = 7 \text{ is a valid solution.} \end{aligned}$$

\]

Determine the possible solutions based on the steps

The algebraic steps shown lead to the factored equation \(0 = (x + 3)(x - 7)\). Based strictly on the algebraic work shown up to that point, the possible solutions (before checking for extraneous solutions) are \(x = -3\) and \(x = 7\).
</reasoning>

<answer>
Study the steps shown to solve the given equation.
\[\sqrt{30-2x} = x-3\]
\[30-2x = x^2-6x+9\]
\[0 = x^2-4x-21\]
\[0 = (x+3)(x-7)\]
Based on the above work, possible solutions of the equation are <blank>\(x = -3\) and \(x = 7\)</blank>.
</answer>

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