If `a ,b ,c` are three distinct positive real numbers, the number of real and distinct roots of `a x^2+2b|x|-c=0` is `0` b. `4` c. `2` d. none of these

Options

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

Explanation

To solve the equation \( ax^2 + 2b|x| - c = 0 \) where \( a, b, c \) are three distinct positive real numbers, we can follow these steps:

### Step 1: Rewrite the Equation
Given the equation:
\[ ax^2 + 2b|x| - c = 0 \]
we can consider two cases for \( |x| \): when \( x \geq 0 \) and when \( x < 0 \).

### Step 2: Case 1: \( x \geq 0 \)
In this case, \( |x| = x \). The equation becomes:
\[ ax^2 + 2bx - c = 0 \]
This is a standard quadratic equation of the form \( Ax^2 + Bx + C = 0 \) where \( A = a \), \( B = 2b \), and \( C = -c \).

### Step 3: Calculate the Discriminant
The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
\[ D = B^2 - 4AC \]
Substituting in our values:
\[ D = (2b)^2 - 4a(-c) = 4b^2 + 4ac = 4(b^2 + ac) \]

### Step 4: Analyze the Discriminant
Since \( a, b, c \) are all positive, \( b^2 + ac > 0 \). Therefore, the discriminant \( D > 0 \), which implies that there are two distinct real roots for the equation \( ax^2 + 2bx - c = 0 \).

### Step 5: Case 2: \( x < 0 \)
In this case, \( |x| = -x \). The equation becomes:
\[ ax^2 - 2bx - c = 0 \]
This is also a quadratic equation of the form \( Ax^2 + Bx + C = 0 \) where \( A = a \), \( B = -2b \), and \( C = -c \).

### Step 6: Calculate the Discriminant for Case 2
The discriminant for this equation is:
\[ D = (-2b)^2 - 4a(-c) = 4b^2 + 4ac = 4(b^2 + ac) \]
Again, since \( a, b, c \) are positive, \( D > 0 \), indicating that there are two distinct real roots for the equation \( ax^2 - 2bx - c = 0 \).

### Step 7: Total Number of Distinct Real Roots
From both cases, we have:
- 2 distinct real roots from \( ax^2 + 2bx - c = 0 \) (for \( x \geq 0 \))
- 2 distinct real roots from \( ax^2 - 2bx - c = 0 \) (for \( x < 0 \))

Thus, the total number of distinct real roots of the original equation \( ax^2 + 2b|x| - c = 0 \) is:
\[ 2 + 2 = 4 \]

### Conclusion
The number of real and distinct roots of the equation \( ax^2 + 2b|x| - c = 0 \) is **4**.