solve (1/4) x squared -3x +6 to be vertex form

To convert the quadratic equation \( \left(\frac{1}{4}\right) x^2 - 3x + 6 \) into vertex form, we need to complete the square. The vertex form of a quadratic equation is given by:

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

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

Here are the steps to convert \( \left(\frac{1}{4}\right) x^2 - 3x + 6 \) to vertex form:

1. **Start with the given quadratic equation:**

\[ y = \left(\frac{1}{4}\right) x^2 - 3x + 6 \]

2. **Factor out the coefficient of \(x^2\) from the first two terms:**

\[ y = \frac{1}{4} (x^2 - 12x) + 6 \]

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

   - Take the coefficient of \(x\), which is \(-12\).
   - Divide it by 2 to get \(-6\).
   - Square it to get \(36\).

4. **Add and subtract this square inside the parentheses:**

\[ y = \frac{1}{4} (x^2 - 12x + 36 - 36) + 6 \]

\[ y = \frac{1}{4} ((x - 6)^2 - 36) + 6 \]

5. **Distribute the \(\frac{1}{4}\) and combine constants:**

\[ y = \frac{1}{4} (x - 6)^2 - \frac{1}{4} \cdot 36 + 6 \]

\[ y = \frac{1}{4} (x - 6)^2 - 9 + 6 \]

\[ y = \frac{1}{4} (x - 6)^2 - 3 \]

So, the vertex form of the given quadratic equation is:

\[ y = \frac{1}{4} (x - 6)^2 - 3 \]

Here, the vertex \((h, k)\) is \((6, -3)\).

Would you like any more details or have any questions? Here are some suggestions:

1. How do you complete the square for the equation \(\left(\frac{1}{4}\right) x^2 - 3x + 6\)?
2. What is the vertex of the quadratic equation \(\left(\frac{1}{4}\right) x^2 - 3x + 6\)?
3. How do you convert a quadratic equation to vertex form?
4. What are the applications of converting a quadratic equation to vertex form?
5. Can you show another example of completing the square to find the vertex form?

Tip: Completing the square is a useful technique not only for converting to vertex form but also for solving quadratic equations and integrating certain expressions.

Learn how to convert the quadratic equation (1/4)x^2 - 3x + 6 to vertex form using the completing the square method. Detailed step-by-step solution with key formulas and concepts covered. Suitable for high school students.

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