Find x, if `x^2 + x = x^3 - x` :

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the equation \( x^2 + x = x^3 - x \), we will follow these steps:

### Step 1: Rearrange the equation
Start by moving all terms to one side of the equation to set it to zero:

\[ x^2 + x - x^3 + x = 0 \]

This simplifies to:

\[ -x^3 + x^2 + 2x = 0 \]

### Step 2: Factor out common terms
Factor out \(-1\) from the entire equation to make the leading coefficient positive:

\[ -x^3 + x^2 + 2x = 0 \implies x^3 - x^2 - 2x = 0 \]

Now, we can factor out \(x\):

\[ x(x^2 - x - 2) = 0 \]

### Step 3: Factor the quadratic
Next, we need to factor the quadratic \(x^2 - x - 2\). We look for two numbers that multiply to \(-2\) and add to \(-1\). These numbers are \(-2\) and \(1\):

\[ x^2 - x - 2 = (x - 2)(x + 1) \]

### Step 4: Write the factored form
Now we can write the entire equation in its factored form:

\[ x(x - 2)(x + 1) = 0 \]

### Step 5: Set each factor to zero
To find the values of \(x\), we set each factor equal to zero:

1. \(x = 0\)
2. \(x - 2 = 0 \implies x = 2\)
3. \(x + 1 = 0 \implies x = -1\)

### Step 6: List the solutions
The solutions to the equation are:

\[ x = 0, \quad x = 2, \quad x = -1 \]

### Conclusion
Thus, the values of \(x\) that satisfy the equation \(x^2 + x = x^3 - x\) are \(0\), \(2\), and \(-1\).

In the context of the options provided, the correct answer is:

\[ \boxed{2} \]

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