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The question asks to write the quadratic expression \(x^2 - 2x - 13\) in the form \((x + c)^2 + d\), and then find the values of \(c\) and \(d\). This process is called completing the square.

Let’s go through the steps:

1. Start with the expression:
   \[
   x^2 - 2x - 13
   \]

2. To complete the square, focus on the first two terms, \(x^2 - 2x\).

3. Take half of the coefficient of \(x\) (which is \(-2\)), divide it by 2, and square it:
   \[
   \left(\frac{-2}{2}\right)^2 = (-1)^2 = 1
   \]

4. Add and subtract this value inside the expression:
   \[
   x^2 - 2x + 1 - 1 - 13
   \]

5. Now, rewrite the first three terms as a perfect square:
   \[
   (x - 1)^2 - 1 - 13
   \]

6. Simplify the constants:
   \[
   (x - 1)^2 - 14
   \]

Now, we have the expression in the form \((x + c)^2 + d\).

- \(c = -1\)
- \(d = -14\)

Would you like a more detailed explanation or have any questions? 

Here are 5 related questions to deepen your understanding:
1. How does completing the square help in solving quadratic equations?
2. What is the geometric interpretation of completing the square?
3. Can you express \(x^2 + 4x + 3\) in completed square form?
4. How is completing the square used in deriving the quadratic formula?
5. What is the difference between factoring and completing the square?

**Tip:** Completing the square is particularly useful when solving quadratic equations that do not factor easily or when rewriting quadratics in vertex form.

Write x^2 - 2x - 13 in the form (x + c)^2 + d. What are the values of c and d?

This tutorial explains how to write the quadratic expression x^2 - 2x - 13 in the form (x + c)^2 + d by completing the square. Includes step-by-step instructions to find the values of c and d. Suitable for Grades 9-11.

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