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To solve the equation \(-2x = x^2 - 6\) by completing the square, we will follow these steps:

### Step 1: Rewrite the equation
First, rearrange the equation so that all terms are on one side:
\[ x^2 + 2x - 6 = 0 \]

### Step 2: Complete the square
To complete the square, we add and subtract the square of half the coefficient of \(x\):
\[ x^2 + 2x + 1 - 1 - 6 = 0 \]
\[ (x + 1)^2 - 7 = 0 \]

Thus, the equation in the form \((x + c)^2 = d\) is:
\[ (x + 1)^2 = 7 \]

### Step 3: Solve for \(x\)
Now, we solve for \(x\) by taking the square root of both sides:
\[ x + 1 = \pm \sqrt{7} \]

This gives us two solutions:
\[ x + 1 = \sqrt{7} \quad \Rightarrow \quad x = \sqrt{7} - 1 \]
\[ x + 1 = -\sqrt{7} \quad \Rightarrow \quad x = -\sqrt{7} - 1 \]

Therefore, the solutions to the equation are:
\[ x = -1 \pm \sqrt{7} \]

### Summary

1. The equation rewritten by completing the square is: \((x + 1)^2 = 7\)
2. The solutions to the equation are: \(x = -1 \pm \sqrt{7}\)

The correct answer is:
\[ \boxed{B \quad x = -1 \pm \sqrt{7}} \]

Would you like more details or have any questions?

Here are 5 more questions you might consider asking:

1. How do you determine the discriminant of a quadratic equation?
2. Can you explain how to graph a quadratic function?
3. What is the significance of the vertex in a parabola?
4. How do you solve quadratic equations using the quadratic formula?
5. Can you show examples of word problems involving quadratic equations?

Tip: When solving equations by completing the square, always ensure you correctly balance the equation by adding and subtracting the same value.

Learn how to solve the equation -2x = x^2 - 6 using the completing the square method with a detailed step-by-step solution. This problem covers key algebraic concepts including quadratic equations, completing the square, and square root calculations. Suitable for high school students.

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