The Quadratic Formula: Factoring Out the Greatest Common Factor (GCF)

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:
$x = 0, -1, 2$.

To solve the given cubic equation $x^3 - 7x^2 + 6x = 0$, follow these steps:

• Step 1: Identify that the equation can be factored by its Greatest Common Factor (GCF).

There is an $x$ common in all terms: $x(x^2 - 7x + 6) = 0$

• Step 2: Factor the quadratic expression $x^2 - 7x + 6$.

Look for two numbers that multiply to $6$ (the constant term) and add up to $-7$ (the coefficient of the linear term). The numbers are $-1$ and $-6$. Thus:

$x^2 - 7x + 6 = (x - 1)(x - 6)$

• Step 3: Set each factor equal to zero to solve for $x$.

Now that the equation is fully factored as $x(x - 1)(x - 6) = 0$, apply the zero product property:

$x = 0$, $x - 1 = 0$ (so $x = 1$), $x - 6 = 0$ (so $x = 6$)

Thus, the solutions to the equation $x^3 - 7x^2 + 6x = 0$ are $x = 0$, $x = 1$, and $x = 6$.

To solve the equation $x^3 + x^2 - 12x = 0$, follow these steps:

• Step 1: Factor out the greatest common factor. The common factor here is $x$.
• Step 2: The equation becomes $x(x^2 + x - 12) = 0$.
• Step 3: Apply the zero-product property. This gives us two equations to solve: $x = 0$ and $x^2 + x - 12 = 0$.
• Step 4: Solve $x = 0$. This is a straightforward solution: $x = 0$.
• Step 5: Solve the quadratic equation $x^2 + x - 12 = 0$. We will factor it:
• Factor as $(x - 3)(x + 4) = 0$.
• Set each factor equal to zero: $x - 3 = 0$ or $x + 4 = 0$.
• Solving these, we obtain $x = 3$ and $x = -4$.

Therefore, the solutions to the equation are $x = 0$, $x = 3$, and $x = -4$.

Thus, the complete solution set for $x$ is $x = 0, 3, -4$.