https://file.aiquickdraw.com/mathbot/file/1725652279723hqdfgddy.jpg

Let's address the problem in the image step by step.

### Part (a)
You are asked to write the quadratic expression \( x^2 - 10x + 35 \) in the form \( (x - p)^2 + q \).

To do this, we can complete the square.

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

   \[
   x^2 - 10x + 35
   \]

2. **Take half the coefficient of \(x\) (which is -10), square it, and add and subtract that square inside the expression:**

   Half of -10 is -5, and \((-5)^2 = 25\).

   So, we can rewrite the expression as:
   
   \[
   x^2 - 10x + 25 - 25 + 35
   \]

3. **Group the perfect square trinomial:**

   \[
   (x - 5)^2 + 10
   \]

So, the expression \( x^2 - 10x + 35 \) in the form \( (x - p)^2 + q \) is:

\[
(x - 5)^2 + 10
\]

### Part (b)
Now, you are asked to find the maximum value of:

\[
\frac{1}{x^2 - 10x + 35}
\]

From part (a), we know that:

\[
x^2 - 10x + 35 = (x - 5)^2 + 10
\]

Therefore, the expression becomes:

\[
\frac{1}{(x - 5)^2 + 10}
\]

To find the maximum value, we need to minimize the denominator \((x - 5)^2 + 10\).

- The minimum value of \((x - 5)^2\) is 0, which occurs when \(x = 5\).
- Substituting \(x = 5\) into the expression gives the minimum value of the denominator:

  \[
  (x - 5)^2 + 10 = 0 + 10 = 10
  \]

Thus, the maximum value of the function is:

\[
\frac{1}{10}
\]

### Summary of the Answers
- **(a)**: The expression \( x^2 - 10x + 35 \) in the form \( (x - p)^2 + q \) is \( (x - 5)^2 + 10 \).
- **(b)**: The maximum value of \( \frac{1}{x^2 - 10x + 35} \) is \( \frac{1}{10} \).

Would you like a detailed explanation of any of these steps, or do you have any other questions? Here are some additional related questions you might find useful:

1. What is the process of completing the square for a general quadratic expression?
2. How do you find the vertex of a parabola given in standard form?
3. Why is the minimum value of a quadratic expression important when finding maximum values of related functions?
4. Can the maximum value of \( \frac{1}{ax^2 + bx + c} \) be found similarly if \(a \neq 1\)?
5. How would you solve for the minimum or maximum of a quadratic function using calculus?

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

Write x^2 - 10x + 35 in the form (x - p)^2 + q and find the maximum value of 1/(x^2 - 10x + 35).

This problem involves rewriting the quadratic expression x^2 - 10x + 35 using completing the square and finding the maximum value of the related function 1/(x^2 - 10x + 35). It covers key algebraic concepts suitable for students in Grades 9-12.

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