Find the 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
We start with the equation:
\[ x^2 + x = x^3 - x \]
To bring all terms to one side, we can subtract \( x^3 \) and add \( x \) to both sides:
\[ x^2 + x + x - x^3 = 0 \]
This simplifies to:
\[ -x^3 + x^2 + 2x = 0 \]

### Step 2: Factor out the common term
We can factor out \( -1 \) from the equation:
\[ -x^3 + x^2 + 2x = 0 \implies -1(x^3 - x^2 - 2x) = 0 \]
Now we can focus on the equation:
\[ x^3 - x^2 - 2x = 0 \]
Next, we can factor out \( x \):
\[ x(x^2 - x - 2) = 0 \]

### Step 3: Solve for \( x \)
Now we have two factors:
1. \( x = 0 \)
2. \( x^2 - x - 2 = 0 \)

For the quadratic equation \( x^2 - x - 2 = 0 \), we can factor it:
\[ (x - 2)(x + 1) = 0 \]
This gives us two more solutions:
1. \( x - 2 = 0 \implies x = 2 \)
2. \( x + 1 = 0 \implies x = -1 \)

### Step 4: List all solutions
So, the solutions to the equation \( x^2 + x = x^3 - x \) are:
\[ x = 0, \quad x = 2, \quad x = -1 \]

### Final Answer
The values of \( x \) that satisfy the equation are \( x = 0, 2, -1 \).