If `x ^(3) -9x =9-x ^(2),` which of the following CANNOT be the value of x ?

Options

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

Answer

To solve the equation \( x^3 - 9x = 9 - x^2 \) and determine which of the following cannot be the value of \( x \), we will follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms to one side of the equation:
\[ x^3 - 9x + x^2 - 9 = 0 \]

### Step 2: Combine like terms
Rearranging the terms gives:
\[ x^3 + x^2 - 9x - 9 = 0 \]

### Step 3: Factor the equation
We can factor the equation by grouping. First, we can group the first two terms and the last two terms:
\[ (x^3 + x^2) + (-9x - 9) = 0 \]
Factoring out common terms from each group:
\[ x^2(x + 1) - 9(x + 1) = 0 \]
Now, factor out the common factor \((x + 1)\):
\[ (x + 1)(x^2 - 9) = 0 \]

### Step 4: Further factor the quadratic
The quadratic \(x^2 - 9\) can be factored as a difference of squares:
\[ (x + 1)(x - 3)(x + 3) = 0 \]

### Step 5: Set each factor to zero
Now, set each factor equal to zero:
1. \(x + 1 = 0 \Rightarrow x = -1\)
2. \(x - 3 = 0 \Rightarrow x = 3\)
3. \(x + 3 = 0 \Rightarrow x = -3\)

### Step 6: List the possible values of \(x\)
The possible values of \(x\) are:
- \(x = -1\)
- \(x = 3\)
- \(x = -3\)

### Step 7: Identify which value cannot be \(x\)
If the options provided include values such as \(1\), \(2\), or any other number not in the list above, we can conclude that those values cannot be \(x\).

Assuming the options include \(1\), \(2\), \(3\), and \(-1\), the value that cannot be \(x\) is:
- \(1\) (since \(1\) is not among the values we found).

### Conclusion
Thus, the value that cannot be \(x\) is \(1\).

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