Complete set of real values of 'a' for which the equation `x^4-2ax^2+x+a^2 -a=0` has all its roots real

Options

A. `[(3)/(4) ,oo)`
B. `[1.oo)`
C. `[2,oo)`
D. `[0,oo)`

Answer

To find the complete set of real values of \( a \) for which the equation

\[ x^4 - 2ax^2 + x + a^2 - a = 0 \]

has all its roots real, we will follow these steps:

### Step 1: Rewrite the equation
We can consider the equation as a quadratic in terms of \( x^2 \). Let \( y = x^2 \). Then we can rewrite the equation as:

\[ y^2 - 2ay + (x + a^2 - a) = 0 \]

### Step 2: Identify the discriminant
For the quadratic equation \( Ay^2 + By + C = 0 \) to have real roots, the discriminant must be non-negative. The discriminant \( D \) is given by:

\[ D = B^2 - 4AC \]

Here, \( A = 1 \), \( B = -2a \), and \( C = x + a^2 - a \). Thus, the discriminant becomes:

\[ D = (-2a)^2 - 4 \cdot 1 \cdot (x + a^2 - a) \]

### Step 3: Simplify the discriminant
Calculating the discriminant:

\[ D = 4a^2 - 4(x + a^2 - a) \]

Expanding this gives:

\[ D = 4a^2 - 4x - 4a^2 + 4a = 4a - 4x \]

### Step 4: Set the discriminant to be non-negative
For the roots to be real, we need:

\[ 4a - 4x \geq 0 \]

This simplifies to:

\[ a \geq x \]

### Step 5: Analyze the roots
Since \( x \) can take any real value, we need to ensure that the quadratic in \( y \) has real roots for all possible values of \( x \).

### Step 6: Consider the case when \( x = 0 \)
At \( x = 0 \):

\[ a \geq 0 \]

### Step 7: Consider the case when \( x \) approaches infinity
As \( x \) approaches infinity, \( a \) must also be able to accommodate larger values. Thus, we can conclude that:

\[ a \geq 0 \]

### Step 8: Find the critical points
Next, we need to check the conditions for the quadratic \( y^2 - 2ay + (x + a^2 - a) = 0 \) to have real roots. The discriminant must remain non-negative for all \( x \).

### Step 9: Set the conditions for \( a \)
We also need to ensure that the quadratic \( x^2 - x + (1 - a) = 0 \) has real roots. The discriminant for this quadratic must also be non-negative:

\[ (-1)^2 - 4(1 - a) \geq 0 \]

This simplifies to:

\[ 1 - 4 + 4a \geq 0 \implies 4a \geq 3 \implies a \geq \frac{3}{4} \]

### Step 10: Combine conditions
Combining the conditions from steps 6 and 9, we find that:

\[ a \geq \frac{3}{4} \]

### Conclusion
Thus, the complete set of real values of \( a \) for which the equation \( x^4 - 2ax^2 + x + a^2 - a = 0 \) has all its roots real is:

\[ \boxed{[ \frac{3}{4}, \infty )} \]