If a,b,c are positive real numbers, then the number of real roots of the equation `ax^2+b|x|+c` is

Options

A. 2
B. 4
C. 0
D. none of these

Answer

To determine the number of real roots of the equation \( ax^2 + b|x| + c = 0 \), where \( a, b, c \) are positive real numbers, we can follow these steps:

### Step 1: Understand the components of the equation
The equation consists of three parts:
- \( ax^2 \): This is a quadratic term. Since \( a > 0 \), the parabola opens upwards.
- \( b|x| \): The absolute value function \( |x| \) is always non-negative, and since \( b > 0 \), this term is also non-negative.
- \( c \): This is a constant term that is positive.

### Step 2: Analyze the equation
The equation can be rewritten as:
\[ ax^2 + b|x| + c = 0 \]
Since \( a, b, c > 0 \), we can observe that:
- \( ax^2 \) is non-negative for all \( x \).
- \( b|x| \) is non-negative for all \( x \).
- \( c \) is a positive constant.

### Step 3: Evaluate the sum
The left-hand side of the equation \( ax^2 + b|x| + c \) is a sum of three non-negative terms, all of which are positive for any real number \( x \) except at \( x = 0 \):
- At \( x = 0 \): \( a(0)^2 + b|0| + c = c > 0 \)
- For any \( x \neq 0 \): both \( ax^2 \) and \( b|x| \) contribute positive values, thus the entire expression remains positive.

### Step 4: Conclusion on the number of real roots
Since the expression \( ax^2 + b|x| + c \) is always positive for all real \( x \), there are no values of \( x \) that can satisfy the equation \( ax^2 + b|x| + c = 0 \).

Thus, we conclude that the number of real roots of the equation \( ax^2 + b|x| + c = 0 \) is **0**.

### Final Answer:
The number of real roots of the equation \( ax^2 + b|x| + c = 0 \) is **0**.

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