https://r2.erweima.ai/mathbot/image/1725032670945j9l3ka3f.jpg

The problem in the image asks to write the quadratic expression \( x^2 - 4x + 5 \) in the form \( (x + a)^2 + b \), where \( a \) and \( b \) are integers to be found.

### Solution

This requires completing the square:

1. **Start with the quadratic expression:**

   \[
   x^2 - 4x + 5
   \]

2. **Rewrite the expression by grouping the first two terms and adding and subtracting a suitable constant to complete the square:**

   \[
   x^2 - 4x + 5 = \left(x^2 - 4x + \left(\frac{-4}{2}\right)^2\right) - \left(\frac{-4}{2}\right)^2 + 5
   \]

   Here, \(\left(\frac{-4}{2}\right)^2 = (-2)^2 = 4\).

3. **Now, substitute this back into the equation:**

   \[
   x^2 - 4x + 5 = (x - 2)^2 - 4 + 5
   \]

4. **Simplify the expression:**

   \[
   x^2 - 4x + 5 = (x - 2)^2 + 1
   \]

### Final Answer

Thus, the quadratic expression can be written as:

\[
(x - 2)^2 + 1
\]

Here, \( a = -2 \) and \( b = 1 \).

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Would you like more details or have any questions?

Here are 5 related questions:
1. What is the purpose of completing the square in solving quadratic equations?
2. How do you derive the vertex form of a quadratic equation using completing the square?
3. How can you use completing the square to solve a quadratic equation that doesn’t factor easily?
4. How would the form change if the coefficient of \( x^2 \) were not 1?
5. What is the geometric interpretation of completing the square?

**Tip:** Completing the square is a valuable technique not only for rewriting quadratic expressions but also for solving quadratic equations and finding the vertex of a parabola.

Learn how to rewrite the quadratic expression x^2 - 4x + 5 in the form (x - 2)^2 + 1 using the completing the square method. This problem involves key concepts in quadratic equations and completing the square technique. Suitable for high school students in Grades 9-12.

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