The set of real values of a for which the equation `x^(2) = a(x+a)` has its roots greater than a is

Options

A. `(-2, -1//2)`
B. `(-1//2, -1//4)`
C. `(-oo, 0)`
D. none of these

Explanation

To solve the problem, we need to find the set of real values of \( a \) for which the equation

\[ x^2 = a(x + a) \]

has its roots greater than \( a \).

### Step 1: Rearranging the Equation
First, we rearrange the equation:

\[ x^2 - ax - a^2 = 0 \]

This is a standard form of a quadratic equation \( Ax^2 + Bx + C = 0 \), where \( A = 1 \), \( B = -a \), and \( C = -a^2 \).

### Step 2: Conditions for Roots
We need to find the conditions under which both roots of this quadratic equation are greater than \( a \).

### Step 3: Using the Quadratic Formula
The roots of the quadratic equation can be found using the quadratic formula:

\[ x = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \]

Substituting \( A \), \( B \), and \( C \):

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

This simplifies to:

\[ x = \frac{a \pm \sqrt{a^2 + 4a^2}}{2} = \frac{a \pm \sqrt{5a^2}}{2} = \frac{a \pm a\sqrt{5}}{2} \]

Thus, the roots are:

\[ x_1 = \frac{a(1 + \sqrt{5})}{2}, \quad x_2 = \frac{a(1 - \sqrt{5})}{2} \]

### Step 4: Analyzing the Roots
For both roots to be greater than \( a \):

1. \( x_1 > a \)
2. \( x_2 > a \)

#### Condition for \( x_1 > a \):

\[ \frac{a(1 + \sqrt{5})}{2} > a \]

Assuming \( a \neq 0 \) (since if \( a = 0 \), the roots cannot be greater than \( 0 \)), we can divide both sides by \( a \):

\[ \frac{1 + \sqrt{5}}{2} > 1 \]

This is always true since \( \sqrt{5} \approx 2.236 \), thus \( \frac{1 + \sqrt{5}}{2} \approx 1.618 > 1 \).

#### Condition for \( x_2 > a \):

\[ \frac{a(1 - \sqrt{5})}{2} > a \]

Again, assuming \( a \neq 0 \), we divide by \( a \):

\[ \frac{1 - \sqrt{5}}{2} > 1 \]

This is not true because \( 1 - \sqrt{5} < 0 \). Therefore, this condition cannot hold for any real \( a \).

### Conclusion
Since the second condition \( x_2 > a \) cannot be satisfied for any real value of \( a \), we conclude that there are no real values of \( a \) for which both roots are greater than \( a \).

Thus, the solution set is:

\[ \text{None} \]