$a^{2}-2a-4=0$

Amanita, let's solve the equation $a^2 - 2a - 4 = 0$ step by step. Here’s the plan:

Step-by-Step Plan

1. Recognize the Equation Type:
   Notice that this is a quadratic equation, which means it has the form $ax^2 + bx + c = 0$.

2. Identify the Coefficients:
   For our equation, $a^2 - 2a - 4 = 0$, the coefficients are:

   • $a = 1$ (the coefficient of $a^2$)
   • $b = -2$ (the coefficient of $a$)
   • $c = -4$ (the constant term)

3. Apply the Quadratic Formula:
   The quadratic formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

We will substitute our values into this formula to find the solutions for $a$.

4. Calculate the Discriminant:
   The discriminant is the part under the square root: $b^2 - 4ac$. This tells us how many real solutions there are.

5. Find the Solutions:
   Substitute the values into the formula and simplify to get the values of $a$.

Now, let's execute each step.

Step 1: Recognize the Equation Type

We have a quadratic equation: $a^2 - 2a - 4 = 0$.

Step 2: Identify the Coefficients

• $a = 1$
• $b = -2$
• $c = -4$

Step 3: Apply the Quadratic Formula

The formula is:

$a = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot (-4)}}{2 \cdot 1}$

Step 4: Calculate the Discriminant

Let's calculate $(-2)^2 - 4 \cdot 1 \cdot (-4)$.

Step 5: Find the Solutions

Now, let's solve the equation for $a$. Amanita, after following all the steps, here are the solutions to the equation $a^2 - 2a - 4 = 0$:

and

These are the two values of $a$ that make the equation true.