To convert the quadratic equation \(-x^2 + 2x - 6\) into vertex form, we can 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.

### Step 1: Factor out the coefficient of \(x^2\)
Start by factoring out \(-1\) from the \(x^2\) and \(x\) terms.

\[
y = -(x^2 - 2x) - 6
\]

### Step 2: Complete the square
Take the coefficient of \(x\) (which is \(-2\)), divide it by 2, and square it.

\[
\left(\frac{-2}{2}\right)^2 = 1
\]

Add and subtract 1 inside the parentheses to complete the square.

\[
y = -(x^2 - 2x + 1 - 1) - 6
\]

### Step 3: Simplify the expression
Group the perfect square trinomial and simplify.

\[
y = -[(x - 1)^2 - 1] - 6
\]

Distribute the \(-1\).

\[
y = -(x - 1)^2 + 1 - 6
\]

\[
y = -(x - 1)^2 - 5
\]

### Final Answer
The vertex form of \(-x^2 + 2x - 6\) is:

\[
y = -(x - 1)^2 - 5
\]

The vertex is \((1, -5)\).

Do you want any details or have any questions? Here are some related problems for further practice:

1. Convert \(x^2 + 6x + 8\) into vertex form.
2. Find the vertex form of \(2x^2 - 4x + 1\).
3. Convert \(x^2 - 4x + 7\) into vertex form.
4. Find the vertex form of \(-3x^2 + 12x - 9\).
5. Write \(4x^2 - 8x + 5\) in vertex form.

**Tip:** Always remember to factor out the coefficient of \(x^2\) before completing the square.

Learn how to convert the quadratic equation -x^2 + 2x - 6 into vertex form using the method of completing the square. This solution covers key concepts in algebra, such as quadratic equations, vertex form, and completing the square. Suitable for students in Grades 9-11.

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