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To rewrite the equation \( y = x^2 + 10x \) in vertex form, we need to complete the square. The vertex form of a quadratic equation is:

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

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

### Steps to Convert to Vertex Form

1. **Rewrite the equation** in a way that allows us to complete the square:
   \[
   y = x^2 + 10x
   \]

2. **Complete the square**:
   - Take half of the coefficient of \(x\), which is 10, and divide it by 2 to get 5.
   - Square 5 to get \(25\).
   - Add and subtract \(25\) inside the equation.

   So we have:
   \[
   y = x^2 + 10x + 25 - 25
   \]

3. **Rewrite the equation** by factoring the perfect square trinomial:
   \[
   y = (x + 5)^2 - 25
   \]

### Final Answer
The vertex form of the equation is:
\[
y = (x + 5)^2 - 25
\]

Here, the vertex of the parabola is at \((-5, -25)\).

Would you like a detailed explanation of the process, or do you have any questions?

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### Related Questions

1. What is the significance of the vertex in a quadratic function?
2. How do you determine the axis of symmetry from the vertex form?
3. Can all quadratic equations be converted to vertex form by completing the square?
4. How do you determine the direction of a parabola from its vertex form?
5. How does the coefficient of \( x^2 \) affect the shape of the parabola?

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**Tip:** Completing the square is a useful technique not only for rewriting equations in vertex form but also for solving quadratic equations.

Write the following equation in vertex form: y = x^2 + 10x

Learn how to rewrite the equation y = x^2 + 10x in vertex form by completing the square. This step-by-step solution helps students understand key algebraic concepts and the process of finding the vertex of a parabola.

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