If `a ^(2) + 1 = a,` then the value of `a ^(3) ` is

Options

A. 0
B. 1
C. `-1`
D. 2

Answer

To solve the equation \( a^2 + 1 = a \) and find the value of \( a^3 \), we can follow these steps:

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

**Hint:** Rearranging the equation helps isolate \( a^2 \) on one side.

### Step 2: Multiply both sides by \( a \)
Next, multiply both sides of the equation \( a^2 = a - 1 \) by \( a \):
\[ a^3 = a \cdot (a - 1) \]

**Hint:** Multiplying by \( a \) allows us to express \( a^3 \) in terms of \( a \).

### Step 3: Expand the right side
Now, expand the right side:
\[ a^3 = a^2 - a \]

**Hint:** Distributing \( a \) over \( (a - 1) \) gives us the expanded form.

### Step 4: Substitute \( a^2 \)
We already know from Step 1 that \( a^2 = a - 1 \). Substitute this into the equation:
\[ a^3 = (a - 1) - a \]

**Hint:** Substituting known values simplifies the equation.

### Step 5: Simplify the expression
Now simplify the right side:
\[ a^3 = a - 1 - a \]
This simplifies to:
\[ a^3 = -1 \]

**Hint:** Combining like terms will help you reach the final answer.

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