Which of the following equations has maximum number of real roots ?

Options

A. `x^2-|x| - 2= 0`
B. `x^2-2|x|+3=0`
C. `x^2-3|x|+2=0 `
D. `x^2+3|x|+2=0`

Explanation

To determine which of the given equations has the maximum number of real roots, we will analyze each equation step by step.

### Step 1: Analyze the first equation
**Equation A:** \( x^2 - |x| - 2 = 0 \)

We can rewrite this equation by considering the definition of the absolute value:
1. \( |x| = x \) when \( x \geq 0 \)
2. \( |x| = -x \) when \( x < 0 \)

**Case 1:** \( x \geq 0 \)
- The equation becomes \( x^2 - x - 2 = 0 \).
- Factoring gives \( (x - 2)(x + 1) = 0 \).
- Roots are \( x = 2 \) and \( x = -1 \) (only \( x = 2 \) is valid since \( x \geq 0 \)).

**Case 2:** \( x < 0 \)
- The equation becomes \( x^2 + x - 2 = 0 \).
- Factoring gives \( (x - 1)(x + 2) = 0 \).
- Roots are \( x = 1 \) and \( x = -2 \) (only \( x = -2 \) is valid since \( x < 0 \)).

**Total Roots for Equation A:** \( x = 2 \) and \( x = -2 \) → **2 real roots.**

### Step 2: Analyze the second equation
**Equation B:** \( x^2 - |x|^2 - 2|x| + 3 = 0 \)

This can be rewritten as:
- \( |x|^2 = x^2 \), so the equation simplifies to \( x^2 - x^2 - 2|x| + 3 = 0 \) → \( -2|x| + 3 = 0 \).
- Solving gives \( |x| = \frac{3}{2} \).

This means:
- \( x = \frac{3}{2} \) and \( x = -\frac{3}{2} \).

**Total Roots for Equation B:** \( x = \frac{3}{2} \) and \( x = -\frac{3}{2} \) → **2 real roots.**

### Step 3: Analyze the third equation
**Equation C:** \( x^2 - 3|x| + 2 = 0 \)

**Case 1:** \( x \geq 0 \)
- The equation becomes \( x^2 - 3x + 2 = 0 \).
- Factoring gives \( (x - 1)(x - 2) = 0 \).
- Roots are \( x = 1 \) and \( x = 2 \).

**Case 2:** \( x < 0 \)
- The equation becomes \( x^2 + 3(-x) + 2 = 0 \) → \( x^2 + 3x + 2 = 0 \).
- Factoring gives \( (x + 1)(x + 2) = 0 \).
- Roots are \( x = -1 \) and \( x = -2 \).

**Total Roots for Equation C:** \( x = 1, 2, -1, -2 \) → **4 real roots.**

### Step 4: Analyze the fourth equation
**Equation D:** \( |x|^2 + 3|x| + 2 = 0 \)

This can be rewritten as:
- \( |x|^2 + 3|x| + 2 = 0 \).
- Factoring gives \( (|x| + 1)(|x| + 2) = 0 \).
- Roots are \( |x| = -1 \) and \( |x| = -2 \) (both are invalid since absolute values cannot be negative).

**Total Roots for Equation D:** **0 real roots.**

### Conclusion
Now we summarize the number of real roots for each equation:
- Equation A: 2 real roots
- Equation B: 2 real roots
- Equation C: 4 real roots
- Equation D: 0 real roots

The equation with the maximum number of real roots is **Equation C: \( x^2 - 3|x| + 2 = 0 \)** which has **4 real roots.**

### Final Answer
The equation that has the maximum number of real roots is **C: \( x^2 - 3|x| + 2 = 0 \)**.

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