To rewrite the quadratic equation $x^2 - 4x - 4$ in vertex form, we use completing the square.

Steps:

1. Start with the given equation: $x^2 - 4x - 4$

2. Isolate the $x^2$ and $x$-terms: $(x^2 - 4x) - 4$

3. Complete the square:

   • Take half of the coefficient of $x$, square it, and add/subtract it within the parenthesis.
   • Coefficient of $x$ is $-4$. Half of it is $-2$, and squaring it gives $(-2)^2 = 4$.
   • Add and subtract 4 inside the parenthesis: $(x^2 - 4x + 4) - 4 - 4$

4. Simplify the perfect square trinomial:

   • $x^2 - 4x + 4$ becomes $(x - 2)^2$.
   • Rewrite: $(x - 2)^2 - 8$

Final Answer:

The vertex form is: $(x - 2)^2 - 8$

Explanation:

• The vertex of the parabola is $(2, -8)$.
• The vertex form makes it easier to identify the transformations and vertex location.

Would you like further clarifications or examples?

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Related Questions:

1. How can you use vertex form to graph a quadratic equation?
2. What is the axis of symmetry for the equation $x^2 - 4x - 4$?
3. How do you derive the vertex form using differentiation?
4. Can you verify the vertex coordinates using a different method?
5. How does the value of $a$ in vertex form affect the parabola's shape?

Tip:

Always double-check your calculations when completing the square, especially when dividing and squaring coefficients!