put 2x^2 +3x +1 into vertex form only use decimals and basic easy algebra 2 method

To convert \( 2x^2 + 3x + 1 \) into vertex form, we can complete the square. Here’s a step-by-step process:

1. **Start with the equation:**
   \[
   y = 2x^2 + 3x + 1
   \]

2. **Factor out the coefficient of \( x^2 \) (which is 2) from the terms involving \( x \):**
   \[
   y = 2(x^2 + \frac{3}{2}x) + 1
   \]

3. **Complete the square inside the parentheses:**
   - Take half of \( \frac{3}{2} \), square it, and add it inside the parentheses.
   - Half of \( \frac{3}{2} \) is \( \frac{3}{4} \), and \( \left(\frac{3}{4}\right)^2 = \frac{9}{16} \).

   So, add \( \frac{9}{16} \) inside the parentheses, but to keep the equation balanced, we also need to subtract \( 2 \times \frac{9}{16} = \frac{9}{8} \) outside the parentheses.

   \[
   y = 2 \left(x^2 + \frac{3}{2}x + \frac{9}{16}\right) + 1 - \frac{9}{8}
   \]

4. **Rewrite as a perfect square:**
   Inside the parentheses, we now have a perfect square:
   \[
   y = 2 \left(x + \frac{3}{4}\right)^2 + 1 - \frac{9}{8}
   \]

5. **Simplify the constants:**
   \[
   1 - \frac{9}{8} = \frac{8}{8} - \frac{9}{8} = -\frac{1}{8}
   \]

   So the equation becomes:
   \[
   y = 2 \left(x + \frac{3}{4}\right)^2 - \frac{1}{8}
   \]

6. **Final Vertex Form:**
   \[
   y = 2 \left(x + 0.75\right)^2 - 0.125
   \]

The vertex form is:
\[
y = 2(x + 0.75)^2 - 0.125
\]

Let me know if you’d like further details or have any questions.

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Here are some related questions:

1. How do you find the vertex from vertex form?
2. How does completing the square help in converting to vertex form?
3. Why do we need to balance the equation when completing the square?
4. What are the benefits of vertex form over standard form in graphing?
5. How do the constants in vertex form relate to the graph's properties?

**Tip:** When completing the square, always ensure you balance the equation by adjusting for terms added and subtracted.

Step-by-step guide to convert 2x^2 + 3x + 1 into vertex form by completing the square. Learn how to apply algebraic methods to rewrite the equation in vertex form y = a(x - h)^2 + k, using decimal values. Ideal for high school students in Grades 9-11.

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