If the equaion `x^(2) + ax+ b = 0` has distinct real roots and `x^(2) + a|x| +b = 0` has only one real root, then

Options

A. `b = 0 , a gt 0 `
B. b = 0, a `lt` 0
C. ` b gt 0, a lt 0 `
D. `b lt 0, a gt 0 `

Explanation

To solve the problem, we need to analyze the two equations given:

1. The first equation is \( x^2 + ax + b = 0 \), which has distinct real roots.
2. The second equation is \( x^2 + a|x| + b = 0 \), which has only one real root.

### Step 1: Analyze the first equation

For the equation \( x^2 + ax + b = 0 \) to have distinct real roots, the discriminant must be greater than zero. The discriminant \( D \) is given by:

\[ D = a^2 - 4b \]

For distinct real roots, we need:

\[ D > 0 \implies a^2 - 4b > 0 \implies a^2 > 4b \quad \text{(1)} \]

### Step 2: Analyze the second equation

The second equation is \( x^2 + a|x| + b = 0 \). We need to consider two cases based on the value of \( x \) (positive and negative).

#### Case 1: \( x \geq 0 \)

In this case, \( |x| = x \), so the equation becomes:

\[ x^2 + ax + b = 0 \]

This is the same as the first equation, which has distinct real roots.

#### Case 2: \( x < 0 \)

In this case, \( |x| = -x \), so the equation becomes:

\[ x^2 - ax + b = 0 \]

For this equation to have only one real root, the discriminant must be zero:

\[ D = (-a)^2 - 4b = a^2 - 4b = 0 \quad \text{(2)} \]

### Step 3: Combine the results

From equation (1), we have \( a^2 > 4b \). From equation (2), we have \( a^2 = 4b \).

Since \( a^2 \) cannot be both greater than and equal to \( 4b \) simultaneously, we need to consider the implications of these conditions.

### Step 4: Determine the values of \( a \) and \( b \)

From equation (2), we can express \( b \) in terms of \( a \):

\[ b = \frac{a^2}{4} \]

Substituting this into the inequality from equation (1):

\[ a^2 > 4\left(\frac{a^2}{4}\right) \implies a^2 > a^2 \]

This is a contradiction unless \( a^2 \) is exactly equal to \( 4b \). Therefore, we conclude that:

1. \( b = 0 \) (since if \( b \) were positive, the inequality would not hold).
2. \( a \) must be greater than 0 for the second equation to have only one root.

Thus, we have:

\[ b = 0 \quad \text{and} \quad a > 0 \]

### Final Result

The range of values for \( a \) and \( b \) is:

\[ \boxed{(a > 0, b = 0)} \]