Solve `|x|^(2)-|x|+4=2x^(2)-3|x|+1`.

Explanation

To solve the equation \( |x|^2 - |x| + 4 = 2x^2 - 3|x| + 1 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ |x|^2 - |x| + 4 = 2x^2 - 3|x| + 1 \]
We can rewrite \( 2x^2 \) as \( 2|x|^2 \) since \( x^2 = |x|^2 \) for any real number \( x \). Thus, the equation becomes:
\[ |x|^2 - |x| + 4 = 2|x|^2 - 3|x| + 1 \]

### Step 2: Move all terms to one side
Now, we will move all terms to one side of the equation:
\[ |x|^2 - |x| + 4 - 2|x|^2 + 3|x| - 1 = 0 \]
This simplifies to:
\[ -|x|^2 + 2|x| + 3 = 0 \]
Multiplying through by -1 gives:
\[ |x|^2 - 2|x| - 3 = 0 \]

### Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation:
\[ |x|^2 - 2|x| - 3 = 0 \]
This factors to:
\[ (|x| + 1)(|x| - 3) = 0 \]

### Step 4: Solve for |x|
Setting each factor equal to zero gives us:
1. \( |x| + 1 = 0 \) → This has no solution since \( |x| \) cannot be negative.
2. \( |x| - 3 = 0 \) → This gives \( |x| = 3 \).

### Step 5: Find the values of x
From \( |x| = 3 \), we have two cases:
1. \( x = 3 \)
2. \( x = -3 \)

### Final Answer
Thus, the solutions to the equation are:
\[ x = 3 \quad \text{and} \quad x = -3 \]

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