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Let's rewrite both of the given quadratic equations in vertex form, which is expressed as:

\[
y = a(x - h)^2 + k
\]

where \((h, k)\) is the vertex of the parabola.

### Part (a) \( y = x^2 - 8x + 25 \)

To convert this into vertex form, we use the method of **completing the square**.

1. Start with the given equation:

   \[
   y = x^2 - 8x + 25
   \]

2. Focus on the quadratic and linear terms \(x^2 - 8x\). To complete the square, take half of the coefficient of \(x\) (which is \(-8\)), square it, and add and subtract this value inside the equation. Half of \(-8\) is \(-4\), and squaring it gives 16.

   \[
   y = (x^2 - 8x + 16) - 16 + 25
   \]

3. Now simplify:

   \[
   y = (x - 4)^2 + 9
   \]

Thus, the vertex form is:

\[
y = (x - 4)^2 + 9
\]

The vertex is \((4, 9)\).

---

### Part (b) \( y = -3x^2 - 12x - 16 \)

For this equation, we again complete the square, but first, factor out the \(-3\) from the quadratic and linear terms:

1. Factor out \(-3\):

   \[
   y = -3(x^2 + 4x) - 16
   \]

2. Now, complete the square inside the parentheses. Take half of the coefficient of \(x\) (which is 4), square it, and add and subtract this value. Half of 4 is 2, and squaring it gives 4.

   \[
   y = -3(x^2 + 4x + 4 - 4) - 16
   \]

3. This becomes:

   \[
   y = -3((x + 2)^2 - 4) - 16
   \]

4. Expand the expression:

   \[
   y = -3(x + 2)^2 + 12 - 16
   \]

5. Simplify:

   \[
   y = -3(x + 2)^2 - 4
   \]

Thus, the vertex form is:

\[
y = -3(x + 2)^2 - 4
\]

The vertex is \((-2, -4)\).

---

Would you like further details on any part of this process, or have any questions?

Here are 5 related questions to explore:
1. How do you find the vertex from a quadratic equation in standard form?
2. What are the advantages of expressing a quadratic equation in vertex form?
3. How does the value of \(a\) in the vertex form affect the graph of the parabola?
4. Can all quadratic equations be written in vertex form using the method of completing the square?
5. How can we derive the vertex formula \( h = \frac{-b}{2a} \) from the standard quadratic form?

**Tip:** Completing the square is a useful technique not only in converting equations to vertex form but also in solving quadratic equations and understanding conic sections.

Rewrite the following in vertex form: a) y = x^2 - 8x + 25, b) y = -3x^2 - 12x - 16

Learn how to rewrite quadratic equations y = x^2 - 8x + 25 and y = -3x^2 - 12x - 16 into vertex form using the method of completing the square. This step-by-step solution explores key algebraic concepts and is ideal for high school students (Grades 9-12).

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