The Quadratic Formula: Equations with variables on both sides

To solve the problem $x^4 - x^3 = 2x^2$, let's proceed as follows:

• Step 1: Set the equation to zero.
  $x^4 - x^3 - 2x^2 = 0$
• Step 2: Factor out the greatest common factor.
  The common factor among all terms is $x^2$.
  Factoring out $x^2$ gives:
  $x^2(x^2 - x - 2) = 0$
• Step 3: Solve the factors.
  This equation breaks into two factors that can be solved separately:
  • $x^2 = 0$
  • $x^2 - x - 2 = 0$
• Step 4: Solve $x^2 = 0$.
  Since $x^2 = 0$, we get:
  $x = 0$
• Step 5: Solve $x^2 - x - 2 = 0$.
  This can be factored further. We look for two numbers that multiply to $-2$ and add up to $-1$.
  These numbers are $-2$ and $1$, so we factor as:
  $(x - 2)(x + 1) = 0$
• Step 6: Solve the quadratic factors.
  Set each factor equal to zero:
  • $x - 2 = 0 \Rightarrow x = 2$
  • $x + 1 = 0 \Rightarrow x = -1$

The solutions to the equation $x^4 - x^3 = 2x^2$ are $x = -1, 0, 2$.

Therefore, the correct answer is:

$x = -1, 2, 0$

To solve the problem $x^3 = x^2 + 2x$, follow these steps:

• Step 1: Re-arrange the equation to have all terms on one side:
  $x^3 - x^2 - 2x = 0$.
• Step 2: Factor out the greatest common factor (GCF), which is $x$:
  $x(x^2 - x - 2) = 0$.
• Step 3: Factor the quadratic expression $x^2 - x - 2$:
  The factors of $-2$ that add up to $-1$ are $-2$ and $1$. Thus, $x^2 - x - 2 = (x-2)(x+1)$.
• Step 4: Combine the factored terms:
  $x(x-2)(x+1) = 0$.

Each factor can be set to zero to find the solutions:

• $x = 0$.
• $x - 2 = 0$, so $x = 2$.
• $x + 1 = 0$, so $x = -1$.

The solutions to the equation are $x = 0, -1, 2$.

Therefore, the correct choice from the given options is:
$x = 0, -1, 2$.