To rewrite the quadratic equation \( y = 7x^2 - 23x + 20 \) in **vertex form**, we will complete the square. The general vertex form of a quadratic equation is:

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

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

### Steps to complete the square:

1. **Factor out the coefficient of \(x^2\) from the first two terms:**
   
   \[
   y = 7(x^2 - \frac{23}{7}x) + 20
   \]

2. **Complete the square inside the parentheses:**

   To complete the square, take half of the coefficient of \(x\), square it, and add and subtract it. The coefficient of \(x\) is \(-\frac{23}{7}\), so:

   \[
   \left(\frac{-\frac{23}{7}}{2}\right)^2 = \left(\frac{-23}{14}\right)^2 = \frac{529}{196}
   \]

   Now, add and subtract \(\frac{529}{196}\) inside the parentheses:

   \[
   y = 7\left(x^2 - \frac{23}{7}x + \frac{529}{196} - \frac{529}{196}\right) + 20
   \]

3. **Rewrite as a perfect square:**

   Now, the first three terms inside the parentheses form a perfect square trinomial:

   \[
   y = 7\left(\left(x - \frac{23}{14}\right)^2 - \frac{529}{196}\right) + 20
   \]

4. **Simplify the constants:**

   Multiply the constant term by 7 and simplify:

   \[
   y = 7\left(x - \frac{23}{14}\right)^2 - 7 \cdot \frac{529}{196} + 20
   \]

   \[
   y = 7\left(x - \frac{23}{14}\right)^2 - \frac{529}{28} + 20
   \]

   Now, convert 20 to a fraction with a denominator of 28:

   \[
   y = 7\left(x - \frac{23}{14}\right)^2 - \frac{529}{28} + \frac{560}{28}
   \]

   \[
   y = 7\left(x - \frac{23}{14}\right)^2 + \frac{31}{28}
   \]

Thus, the vertex form of the quadratic equation is:

\[
y = 7\left(x - \frac{23}{14}\right)^2 + \frac{31}{28}
\]

### Vertex:
The vertex of the parabola is \(\left(\frac{23}{14}, \frac{31}{28}\right)\).

---

Would you like more details on any step? Here are 5 follow-up questions:

1. How is completing the square used to convert standard form to vertex form?
2. Can you explain the significance of the

Learn how to convert the quadratic equation y=7x^2-23x+20 into vertex form using the method of completing the square. This detailed, step-by-step explanation is ideal for students in Grades 9-11 studying quadratic equations and vertex form transformations.

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