The Quadratic Formula: Missing formula

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:

$x_1 = 0, x_2 = -1$

To solve the quadratic equation $5x^2 + 10x = 0$, we will follow these steps:

• Step 1: Factor the equation: Identify the greatest common factor, which in this case is $5x$.
• Step 2: Apply the zero product property: Set each factor equal to zero and solve for $x$.

Now, let's execute these steps:

Step 1: Factor the equation.
The equation is $5x^2 + 10x = 0$.
Factor out the greatest common factor, $5x$, from both terms:
$5x(x + 2) = 0$.

Step 2: Apply the zero product property.
According to this property, if the product of factors is zero, then at least one of the factors must be zero.
Thus, we set each factor equal to zero:

• $5x = 0$
• $x + 2 = 0$

Solving these, we find:

$5x = 0 \Rightarrow x = 0$

$x + 2 = 0 \Rightarrow x = -2$

Therefore, the solutions to the equation $5x^2 + 10x = 0$ are $x_1 = 0$ and $x_2 = -2$.

The correct choice among the provided options is: $x_1 = 0, x_2 = -2$.

To solve the equation $3x^2 - 9x = 0$, we will factor the quadratic expression:

• Step 1: Factor the quadratic.
  Observe that both terms in $3x^2 - 9x$ have a common factor of $3x$. Thus, we factor by pulling out $3x$:
  $3x(x - 3) = 0$
• Step 2: Apply the zero product property.
  According to the zero product property, if $3x(x - 3) = 0$, then either $3x = 0$ or $x - 3 = 0$ must hold true.
• Step 3: Solve each factor for $x$.
  - For $3x = 0$:
  Divide both sides by 3:
  $x = 0$
• - For $x - 3 = 0$:
  Add 3 to both sides:
  $x = 3$

Therefore, the solutions to the equation are $x_1 = 3$ and $x_2 = 0$.

These solutions correspond to the choice: $x_1=3,x_2=0$.