By completing the square, express the equation xy=4+2x−x2 in the form xy=a(x+b)2+c. Write down the values of a, b, and c. Hence, find the coordinates of the maximum point of the curve.

Let's solve the problem step by step.

### Step 1: Start with the given equation.
The equation is:
\[
xy = 4 + 2x - x^2
\]

### Step 2: Rearrange the right side to prepare for completing the square.
First, write the quadratic expression \( -x^2 + 2x \) in a standard form suitable for completing the square:
\[
xy = -x^2 + 2x + 4
\]

### Step 3: Complete the square.
To complete the square for the quadratic expression \( -x^2 + 2x \), we can factor out \(-1\) from the first two terms:
\[
xy = -(x^2 - 2x) + 4
\]
Now, complete the square inside the parentheses. To complete the square for \( x^2 - 2x \), add and subtract \( \left(\frac{2}{2}\right)^2 = 1 \):
\[
xy = -\left(x^2 - 2x + 1 - 1\right) + 4
\]
This simplifies to:
\[
xy = -\left((x - 1)^2 - 1\right) + 4
\]
Now, distribute the negative sign:
\[
xy = -(x - 1)^2 + 1 + 4
\]
Simplify further:
\[
xy = -(x - 1)^2 + 5
\]

### Step 4: Express the equation in the required form \( xy = a(x + b)^2 + c \).
The equation is now in the form:
\[
xy = -1 \cdot (x - 1)^2 + 5
\]
This can be rewritten as:
\[
xy = -1 \cdot \left(x - (-1)\right)^2 + 5
\]
Therefore, comparing with \( xy = a(x + b)^2 + c \), we have:
- \( a = -1 \)
- \( b = -1 \)
- \( c = 5 \)

### Step 5: Find the coordinates of the maximum point of the curve.
The maximum point occurs where the squared term \( (x - 1)^2 \) is zero, because the negative sign in front of the square means that the expression is maximized when the square is minimized (which is zero).

Set \( (x - 1)^2 = 0 \):
\[
x - 1 = 0 \quad \Rightarrow \quad x = 1
\]

Substitute \( x = 1 \) back into the equation to find \( y \):
\[
xy = -1 \cdot (1 - 1)^2 + 5 = 5
\]
Since \( x = 1 \), we have:
\[
y = \frac{5}{1} = 5
\]

So, the coordinates of the maximum point are \( (1, 5) \).

### Final Answer:
- **a**: \(-1\)
- **b**: \(-1\)
- **c**: \(5\)
- **Maximum point**: \((1, 5)\)

Would you like more details on this method or have any other questions?

### Related Questions:
1. What is the process of completing the square for a general quadratic expression?
2. How does the sign of \( a \) in \( y = a(x + b)^2 + c \) affect the direction of the parabola?
3. Can you express the maximum or minimum point of a parabola using the vertex form equation?
4. How do you derive the vertex form of a quadratic function from its standard form?
5. How does the transformation \( (x - h, y - k) \) shift the graph of a function?

### Tip:
When completing the square, always ensure that the coefficient of \( x^2 \) is 1. If not, factor it out first before completing the square.

Learn how to complete the square to solve the equation xy = 4 + 2x - x^2. Find the values of a, b, and c in the form xy = a(x + b)^2 + c and determine the coordinates of the maximum point of the curve. Suitable for advanced high school students.

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