The real roots of the equation `|x|^3-3x^2+3|x|-2=0` are

Options

A. 0, 2
B. `+- 1`
C. `+- 2`
D. 1, 2

Answer

To find the real roots of the equation \( |x|^3 - 3x^2 + 3|x| - 2 = 0 \), we can follow these steps:

### Step 1: Substitute \( |x| \)
Let \( y = |x| \). Then, the equation becomes:
\[ y^3 - 3y^2 + 3y - 2 = 0 \]

### Step 2: Factor the cubic equation
We can try to find the roots of the cubic equation. By inspection or using the Rational Root Theorem, we can test \( y = 2 \):
\[ 2^3 - 3(2^2) + 3(2) - 2 = 8 - 12 + 6 - 2 = 0 \]
Thus, \( y = 2 \) is a root.

### Step 3: Factor out \( (y - 2) \)
Now, we can factor the cubic polynomial using synthetic division or polynomial long division:
\[ y^3 - 3y^2 + 3y - 2 = (y - 2)(y^2 - y + 1) \]

### Step 4: Solve the quadratic equation
Next, we need to solve the quadratic equation \( y^2 - y + 1 = 0 \). We can find the discriminant \( D \):
\[ D = b^2 - 4ac = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]
Since \( D < 0 \), this quadratic has no real roots.

### Step 5: Find the value of \( y \)
The only real solution we have is from \( y - 2 = 0 \):
\[ y = 2 \]

### Step 6: Relate back to \( x \)
Since \( y = |x| \), we have:
\[ |x| = 2 \]
This gives us two possible values for \( x \):
\[ x = 2 \quad \text{or} \quad x = -2 \]

### Final Answer
The real roots of the equation \( |x|^3 - 3x^2 + 3|x| - 2 = 0 \) are:
\[ x = 2 \quad \text{and} \quad x = -2 \]

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