If a,b,c are positive real numbers, then the number of real roots of the equation `ax^(2)+b|x|+c=0` 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: Rewrite the equation
The equation can be rewritten as:
\[ a|x|^2 + b|x| + c = 0 \]
This is a quadratic equation in terms of \( |x| \).

### Step 2: Identify the coefficients
Let \( y = |x| \). Then the equation becomes:
\[ ay^2 + by + c = 0 \]
where \( a, b, c > 0 \).

### Step 3: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( Ay^2 + By + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
In our case:
\[ D = b^2 - 4ac \]

### Step 4: Analyze the discriminant
Since \( a, b, c \) are all positive, we can analyze the discriminant:
- If \( D < 0 \), the quadratic equation has no real roots.
- If \( D = 0 \), the quadratic equation has exactly one real root.
- If \( D > 0 \), the quadratic equation has two distinct real roots.

### Step 5: Determine the nature of the roots
Since \( a, b, c > 0 \), we can see that:
\[ b^2 - 4ac < 0 \]
This is because \( b^2 \) is always less than \( 4ac \) when \( a, b, c \) are positive real numbers.

### Conclusion
Since the discriminant \( D < 0 \), the quadratic equation \( ay^2 + by + c = 0 \) has no real roots. Therefore, the original equation \( ax^2 + b|x| + c = 0 \) also has no real roots.

Thus, the number of real roots of the equation \( ax^2 + b|x| + c = 0 \) is:
\[ \text{Number of real roots} = 0 \]