Which statement best reflects the solution(s) of the equation?

There is only one solution: .
The solution is an extraneous solution.
There are two solutions: and .
There is only one solution: .
The solution is an extraneous solution.
There is only one solution: .
The solution is an extraneous solution.

Ask by Carlson Logan. in the United States

Dec 20,2024

Question

Which statement best reflects the solution(s) of the equation?

There is only one solution: .
The solution is an extraneous solution.
There are two solutions: and .
There is only one solution: .
The solution is an extraneous solution.
There is only one solution: .
The solution is an extraneous solution.

Ask by Carlson Logan. in the United States

Dec 20,2024

UpStudy AI · Accepted Answer

To determine the solutions of the equation:

$$
\sqrt{3x + 1} - x + 3 = 0
$$

**Step 1: Isolate the Square Root**

$$
\sqrt{3x + 1} = x - 3
$$

**Step 2: Determine the Domain**

- The expression inside the square root must be non-negative:
  $$
  3x + 1 \geq 0 \implies x \geq -\frac{1}{3}
  $$
- The right side of the equation ($$x - 3$$) must also be non-negative:
  $$
  x - 3 \geq 0 \implies x \geq 3
  $$
  
**Therefore, any potential solution must satisfy $$x \geq 3$$.**

**Step 3: Square Both Sides**

$$
3x + 1 = (x - 3)^2
$$

$$
3x + 1 = x^2 - 6x + 9
$$

$$
0 = x^2 - 9x + 8
$$

**Step 4: Solve the Quadratic Equation**

$$
x^2 - 9x + 8 = 0
$$

$$
(x - 1)(x - 8) = 0
$$

$$
x = 1 \quad \text{or} \quad x = 8
$$

**Step 5: Verify the Solutions Against the Domain**

- **$$x = 1$$:**
  - Does not satisfy $$x \geq 3$$.
  - **Extraneous Solution.**
  
- **$$x = 8$$:**
  - Satisfies $$x \geq 3$$.
  - **Valid Solution.**

**Conclusion:**

- **Only $$x = 8$$ is a valid solution.**
- **$$x = 1$$ is an extraneous solution.**

**Best Statement:**

**There is only one solution: $$x = 8$$.**
There is only one solution: $$ x = 8 $$.

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