Given: The equation is x^2 = x - 1.
To Find: The value of (α^2/β) - (β^2/α).

Explanation:
Let's solve the given equation:
x^2 = x - 1

Rearranging the equation, we get:
x^2 - x + 1 = 0

We can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = -1, and c = 1.

Substituting the values, we get:
x = (-(-1) ± √((-1)^2 - 4(1)(1))) / (2(1))
x = (1 ± √(1 - 4)) / 2
x = (1 ± √(-3)) / 2

Since the square root of a negative number is not a real number, we can conclude that the given equation has no real roots.

However, we can find the complex roots of the equation.

Complex Roots:
The complex roots of a quadratic equation can be written in the form α = p + qi and β = p - qi, where p and q are real numbers and i is the imaginary unit (√(-1)).

In this case, since the equation has no real roots, both α and β will be complex numbers.

Using Vieta's Formulas:
Vieta's formulas state that for a quadratic equation ax^2 + bx + c = 0 with roots α and β, the sum of the roots is α + β = -b/a and the product of the roots is αβ = c/a.

In this case, a = 1, b = -1, and c = 1.

Using the sum of roots formula, we get:
α + β = -(-1)/1
α + β = 1

Using the product of roots formula, we get:
αβ = 1/1
αβ = 1

Therefore, the sum of the roots α and β is 1 and the product of the roots is 1.

Calculating α^2/β and β^2/α:
To find the value of (α^2/β) - (β^2/α), we need to calculate α^2/β and β^2/α separately.

α^2/β = (α/β) * α = α(α/β) = α^2 * (1/β) = α^2/β
Similarly, β^2/α = β(β/α) = β^2 * (1/α) = β^2/α

Therefore, (α^2/β) - (β^2/α) = α^2/β - β^2/α

Substituting the Values:
Since we know that α + β = 1 and αβ = 1, we can substitute these values into the expression (α^2/β) - (β^2/α).

Using the identity (a + b)^2 = a^2 + b^2 + 2