To convert a quadratic equation from **general form** to **standard form**, it's important to recognize what each form looks like.

### 1. **General form** of a quadratic equation:
\[
ax^2 + bx + c = 0
\]
Where:
- \(a\), \(b\), and \(c\) are constants.
- \(x\) is the variable.

### 2. **Standard form** (vertex form) of a quadratic equation:
\[
y = a(x - h)^2 + k
\]
Where:
- \(a\) is the same as in the general form.
- \((h, k)\) is the vertex of the parabola.

### **Steps to convert** general form to standard form:

#### Step 1: Begin with the general form
\[
ax^2 + bx + c = 0
\]

#### Step 2: Factor out the \(a\) if necessary (if \(a \neq 1\)):
\[
a(x^2 + \frac{b}{a}x) + c = 0
\]

#### Step 3: Complete the square
To complete the square, take the coefficient of \(x\) (which is \(\frac{b}{a}\)), divide it by 2, then square it:
\[
\left(\frac{\frac{b}{a}}{2}\right)^2 = \frac{b^2}{4a^2}
\]
Now, add and subtract this value inside the parentheses:
\[
a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right) + c = 0
\]

#### Step 4: Rearrange terms to form a perfect square trinomial
The terms inside the parentheses form a perfect square trinomial:
\[
a\left[\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right] + c = 0
\]

#### Step 5: Simplify the constant term
Simplify the equation by factoring and combining constants:
\[
y = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c
\]
Where \(h = -\frac{b}{2a}\) and \(k = c - \frac{b^2}{4a}\).

The final equation is now in **standard form**:
\[
y = a(x - h)^2 + k
\]

### Example:
Convert \(x^2 + 6x + 8 = 0\) to standard form.

#### Step 1: Complete the square
- Start with \(x^2 + 6x\).
- Take \(6\), divide by 2, and square it: \(\left(\frac{6}{2}\right)^2 = 9\).
- Add and subtract 9: \((x^2 + 6x + 9 - 9) + 8 = 0\).
- Rearrange: \((x + 3)^2 - 9 + 8 = 0\).

#### Step 2: Simplify
\[
(x + 3)^2 - 1 = 0
\]
So, the standard form is \(y = (x + 3)^2 - 1\).

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### Relative questions:
1. How do you handle cases where \(a \neq 1\) when converting?
2. Can the general form be derived back from the standard form?
3. What are the advantages of the standard form in solving quadratic equations?
4. How does completing the square help in understanding the geometry of a parabola?
5. Can this process be applied to non-quadratic polynomials?

### Tip:
Always check your work by expanding the standard form back into the general form to confirm correctness.

Learn how to convert quadratic equations from general form (ax^2 + bx + c = 0) to standard form (y = a(x - h)^2 + k) using the completing the square method. This detailed step-by-step guide is perfect for high school students studying algebra and quadratic functions.

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