The equation `x^(2) + ax + b^(2) = 0` has two roots each of which exceeds a member c, then

Options

A. `a^(2) lt 4b^(2)`
B. `c^(2) + ac + b^(2) gt 0`
C. `(-a)/(2) lt c `
D. None of these

Answer

To solve the problem, we need to analyze the quadratic equation given by:

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

We are told that this equation has two roots, say \( \alpha \) and \( \beta \), and both roots exceed a certain number \( c \). Our goal is to find a condition that must hold true given this information.

### Step 1: Understanding the Roots
The roots of the quadratic equation can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In our case, \( a = 1 \), \( b = a \), and \( c = b^2 \). Thus, the roots are:

\[ x = \frac{-a \pm \sqrt{a^2 - 4(1)(b^2)}}{2(1)} \]

This simplifies to:

\[ x = \frac{-a \pm \sqrt{a^2 - 4b^2}}{2} \]

### Step 2: Condition for Roots to Exceed \( c \)
Since both roots \( \alpha \) and \( \beta \) exceed \( c \), we can express this condition mathematically as:

\[ \alpha > c \quad \text{and} \quad \beta > c \]

This implies that both roots are greater than \( c \).

### Step 3: Setting Up the Inequality
To express the condition that both roots exceed \( c \), we can set up the following inequalities:

1. \( \alpha > c \)
2. \( \beta > c \)

This can be rewritten in terms of the quadratic function evaluated at \( c \):

\[ f(c) = c^2 + ac + b^2 \]

For both roots to be greater than \( c \), the value of the quadratic function at \( c \) must be positive:

\[ f(c) > 0 \]

### Step 4: Conclusion
Thus, we can conclude that:

\[ c^2 + ac + b^2 > 0 \]

This condition must hold true for both roots of the quadratic equation to exceed the number \( c \).

### Final Condition
The final condition derived from the analysis is:

\[ c^2 + ac + b^2 > 0 \]