Quadratic Equation Basics: Solving x² + x = 0

To solve the equation $x^2 + x = 0$, we will use a step-by-step method:

• Step 1: Identify the equation form.
  The equation given is $x^2 + x = 0$. This is a quadratic equation in a simpler form since it can be factored easily.
• Step 2: Factor the equation.
  We notice that both terms of the equation have a common factor, which is $x$. Therefore, we can factor out $x$ as follows:

$x(x + 1) = 0$

• Step 3: Apply the zero product property.
  The zero product property tells us that if the product of two factors is zero, then at least one of the factors must be zero. Thus, set each factor equal to zero:

$x = 0$
$x + 1 = 0$

• Step 4: Solve each equation.
  The first equation gives $x = 0$ directly. The second equation can be solved by subtracting 1 from both sides to find:

$x = -1$

Therefore, the solutions to the equation $x^2 + x = 0$ are $x_1 = 0$ and $x_2 = -1$.

Therefore, the correct answer is:

$x_1 = 0, x_2 = -1$