write this expression in the form of (x+p)^2+q or q-(x+p)^2

1. x^2+2x+5

2. x^2+20x-45

3.3-6x-x^2


To rewrite each expression in the form \((x + p)^2 + q\) or \(q - (x + p)^2\), we'll complete the square.

### 1. \(x^2 + 2x + 5\)

1. Start with the expression: \(x^2 + 2x + 5\).
2. To complete the square, take the coefficient of \(x\) (which is 2), divide it by 2, and then square it:
   \[
   \left(\frac{2}{2}\right)^2 = 1
   \]
3. Rewrite the expression by adding and subtracting this value:
   \[
   x^2 + 2x + 1 - 1 + 5 = (x + 1)^2 + 4
   \]
4. The expression in the desired form is:
   \[
   (x + 1)^2 + 4
   \]

### 2. \(x^2 + 20x - 45\)

1. Start with the expression: \(x^2 + 20x - 45\).
2. To complete the square, take the coefficient of \(x\) (which is 20), divide it by 2, and then square it:
   \[
   \left(\frac{20}{2}\right)^2 = 100
   \]
3. Rewrite the expression by adding and subtracting this value:
   \[
   x^2 + 20x + 100 - 100 - 45 = (x + 10)^2 - 145
   \]
4. The expression in the desired form is:
   \[
   (x + 10)^2 - 145
   \]

### 3. \(3 - 6x - x^2\)

1. Start with the expression: \(3 - 6x - x^2\).
2. First, rearrange the terms and factor out \(-1\) to make the \(x^2\) term positive:
   \[
   -(x^2 + 6x) + 3
   \]
3. Complete the square inside the parentheses. Take the coefficient of \(x\) (which is 6), divide it by 2, and then square it:
   \[
   \left(\frac{6}{2}\right)^2 = 9
   \]
4. Rewrite the expression by adding and subtracting this value inside the parentheses:
   \[
   -(x^2 + 6x + 9 - 9) + 3 = -(x + 3)^2 + 12
   \]
5. The expression in the desired form is:
   \[
   12 - (x + 3)^2
   \]

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Would you like any further details or have any questions about the steps? 

Here are five related questions to deepen your understanding:

1. How do you complete the square for a quadratic expression with a leading coefficient other than 1?
2. Can you rewrite \(x^2 - 4x + 7\) in the form \((x + p)^2 + q\)?
3. How does completing the square help in solving quadratic equations?
4. What is the significance of the vertex form \((x + p)^2 + q\) of a quadratic equation?
5. How would the process change if you were asked to factor instead of completing the square?

**Tip:** Completing the square is particularly useful for converting quadratic equations into vertex form, which makes it easier to identify the maximum or minimum value of the function.

write this expression in the form of (x+p)^2+q or q-(x+p)^2

1. x^2+2x+5

2. x^2+20x-45

3. 3-6x-x^2

Explore how to rewrite quadratic expressions like x^2 + 2x + 5 in the form (x + p)^2 + q or q - (x + p)^2 by completing the square. This guide is ideal for students in Grades 9-11 looking to understand the vertex form of quadratic equations.

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