If `a^2 + 1 = a`, then the value of `a^12 + a^6 + 1` is :

Options

A. `-3`
B. 1
C. 2
D. 3

Answer

To solve the equation \( a^2 + 1 = a \) and find the value of \( a^{12} + a^6 + 1 \), we can follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ a^2 + 1 = a \]
Rearranging gives:
\[ a^2 - a + 1 = 0 \]

### Step 2: Solve the quadratic equation
We can use the quadratic formula to solve for \( a \):
\[ a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = -1, c = 1 \):
\[ a = \frac{1 \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{1 \pm \sqrt{1 - 4}}{2} = \frac{1 \pm \sqrt{-3}}{2} \]
This gives us:
\[ a = \frac{1 \pm i\sqrt{3}}{2} \]

### Step 3: Find \( a^3 \)
To find \( a^{12} + a^6 + 1 \), we first need to find \( a^3 \). We can use the original equation:
\[ a^2 = a - 1 \]
Multiplying both sides by \( a \):
\[ a^3 = a \cdot a^2 = a(a - 1) = a^2 - a = (a - 1) - a = -1 \]
Thus, we have:
\[ a^3 = -1 \]

### Step 4: Calculate \( a^6 \) and \( a^{12} \)
Using \( a^3 = -1 \):
\[ a^6 = (a^3)^2 = (-1)^2 = 1 \]
\[ a^{12} = (a^6)^2 = 1^2 = 1 \]

### Step 5: Substitute into the expression
Now we can substitute \( a^{12} \) and \( a^6 \) into the expression \( a^{12} + a^6 + 1 \):
\[ a^{12} + a^6 + 1 = 1 + 1 + 1 = 3 \]

### Conclusion
Thus, the value of \( a^{12} + a^6 + 1 \) is:
\[ \boxed{3} \]