https://r2.erweima.ai/mathbot/image/1727464987429sc95l9hu.jpg

The equation provided in the image is:

\[
\sqrt{x^2 - 4x + 3} - 1 + \frac{1}{x} \left( \sqrt{4x - x^2 - 3 + 1} \right) = 0
\]

Let's simplify this step by step.

### Step 1: Simplify each square root term

1. The first square root term: 
   \[
   \sqrt{x^2 - 4x + 3}
   \]
   This can be factored:
   \[
   x^2 - 4x + 3 = (x - 3)(x - 1)
   \]
   Thus, the square root simplifies to:
   \[
   \sqrt{(x - 3)(x - 1)}
   \]

2. The second square root term:
   \[
   \sqrt{4x - x^2 - 3 + 1} = \sqrt{4x - x^2 - 2}
   \]
   This is the simplified form of the second square root term.

### Step 2: Rewrite the equation
The equation becomes:
\[
\sqrt{(x - 3)(x - 1)} - 1 + \frac{1}{x} \cdot \sqrt{4x - x^2 - 2} = 0
\]

### Step 3: Trial for specific values of \(x\)
To solve the equation, we can try possible values for \(x\).

Let me calculate this.The solutions to the equation are:

1. \( x = 1 \)
2. \( x = 2 - \sqrt{2} \)
3. \( x = 2 + \sqrt{2} \)

Additionally, there is a complex root expressed as \( \text{CRootOf}(x^5 - 3x^4 + x^3 - 7x^2 + 2x - 2, 0) \), which requires further numerical methods to evaluate precisely.

Among these, the most straightforward real solution is \( x = 1 \).

Would you like further details on any of these solutions, or have any other questions?

Here are 5 related questions:
1. How does factoring help simplify square root expressions in algebra?
2. What other values of \(x\) could be solutions based on trial methods?
3. How can complex roots be calculated in higher-order polynomials?
4. What is the significance of testing integer values of \(x\) in such equations?
5. Can numerical methods be used to approximate non-integer roots of polynomial equations?

**Tip**: Always check for both real and complex roots when solving algebraic equations involving radicals or higher powers.

Корнем уравнения \( \sqrt{x^2 - 4x + 3} - 1 + \frac{1}{x} \sqrt{4x - x^2 - 3 + 1} = 0 \) является число?

Learn how to solve the algebraic equation involving square roots and factoring: \( \sqrt{x^2 - 4x + 3} - 1 + \frac{1}{x} \sqrt{4x - x^2 - 3 + 1} = 0 \). Step-by-step explanation with detailed factoring and simplification. Suitable for students in Grades 9-11.

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