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To solve the given problem, we need to rewrite the quadratic expression \( x^2 - 14x - 1 \) in the form \( (x + a)^2 + b \), where \( a \) and \( b \) are integers.

### Step 1: Completing the Square
To rewrite the expression \( x^2 - 14x - 1 \) in the form \( (x + a)^2 + b \), we first complete the square.

1. Start with the quadratic expression:
   \[
   x^2 - 14x - 1
   \]

2. Focus on the first two terms \( x^2 - 14x \) and complete the square. Take half of the coefficient of \( x \), which is \(-14\), and square it:
   \[
   \left(\frac{-14}{2}\right)^2 = (-7)^2 = 49
   \]

3. Add and subtract this square inside the expression to complete the square:
   \[
   x^2 - 14x + 49 - 49 - 1
   \]

4. Group the perfect square trinomial and combine the constants:
   \[
   (x - 7)^2 - 50
   \]

### Step 2: Rewrite in the Desired Form
The expression is now in the form \( (x + a)^2 + b \), where \( a = -7 \) and \( b = -50 \).

Thus, the expression \( x^2 - 14x - 1 \) can be rewritten as:
\[
(x - 7)^2 - 50
\]

### Final Answer
- \( a = -7 \)
- \( b = -50 \)

Would you like more details or have any questions?

Here are some related questions you might find helpful:
1. How do you complete the square for a quadratic expression in general?
2. What is the geometric interpretation of completing the square?
3. How can completing the square help in solving quadratic equations?
4. Can you convert a quadratic expression to vertex form using a different method?
5. How do you determine if a quadratic expression has real or complex roots?

**Tip:** Completing the square is a valuable technique for solving quadratics and understanding their properties, including finding the vertex of a parabola.

Write x^2 - 14x - 1 in the form (x + a)^2 + b where a and b are integers to be found.

Learn how to rewrite the quadratic expression x^2 - 14x - 1 in the form (x + a)^2 + b using the completing the square method. This step-by-step guide covers key algebraic techniques suitable for students in Grades 9-10.

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