How many integer pairs (x,y) satisfy `x^(2) + 4y^(2) -2xy -2x - 4y -8=0` ?

Answer

To find the integer pairs \((x, y)\) that satisfy the equation

\[ x^2 + 4y^2 - 2xy - 2x - 4y - 8 = 0, \]

we can start by rearranging and simplifying the equation.

### Step 1: Rewrite the equation

We can rearrange the equation to group terms:

\[ x^2 - 2xy - 2x + 4y^2 - 4y - 8 = 0. \]

### Step 2: Complete the square

Next, we can complete the square for the terms involving \(x\) and \(y\).

1. For \(x\):
\[ x^2 - 2xy - 2x = (x - y - 1)^2 - (y + 1)^2. \]

2. For \(y\):
\[ 4y^2 - 4y = 4(y^2 - y) = 4\left((y - \frac{1}{2})^2 - \frac{1}{4}\right) = 4(y - \frac{1}{2})^2 - 1. \]

Putting it all together, we have:

\[ (x - y - 1)^2 + 4(y - \frac{1}{2})^2 - 1 - 8 = 0. \]

This simplifies to:

\[ (x - y - 1)^2 + 4(y - \frac{1}{2})^2 = 9. \]

### Step 3: Set up the new equation

Now we can express this as:

\[ (x - y - 1)^2 + 4(y - \frac{1}{2})^2 = 9. \]

Let \(a = x - y - 1\) and \(b = 2(y - \frac{1}{2})\). Then we have:

\[ a^2 + b^2 = 9. \]

### Step 4: Find integer solutions

The equation \(a^2 + b^2 = 9\) has integer solutions. The possible pairs \((a, b)\) that satisfy this equation are:

1. \((3, 0)\)
2. \((0, 3)\)
3. \((-3, 0)\)
4. \((0, -3)\)
5. \((2, \pm 2)\)
6. \((\pm 2, 2)\)

### Step 5: Back-substitute to find \((x, y)\)

Now we can back-substitute to find the corresponding \((x, y)\) pairs for each \((a, b)\):

1. For \(a = 3\), \(b = 0\):
- \(x - y - 1 = 3 \Rightarrow x - y = 4\)
- \(2(y - \frac{1}{2}) = 0 \Rightarrow y = \frac{1}{2}\) (not an integer)

2. For \(a = 0\), \(b = 3\):
- \(x - y - 1 = 0 \Rightarrow x - y = 1\)
- \(2(y - \frac{1}{2}) = 3 \Rightarrow y = 3\)
- \(x = 4\) → \((4, 3)\)

3. For \(a = -3\), \(b = 0\):
- \(x - y - 1 = -3 \Rightarrow x - y = -2\)
- \(2(y - \frac{1}{2}) = 0 \Rightarrow y = \frac{1}{2}\) (not an integer)

4. For \(a = 0\), \(b = -3\):
- \(x - y - 1 = 0 \Rightarrow x - y = 1\)
- \(2(y - \frac{1}{2}) = -3 \Rightarrow y = -1\)
- \(x = 0\) → \((0, -1)\)

5. For \(a = 2\), \(b = 2\):
- \(x - y - 1 = 2 \Rightarrow x - y = 3\)
- \(2(y - \frac{1}{2}) = 2 \Rightarrow y = 1\)
- \(x = 4\) → \((4, 1)\)

6. For \(a = 2\), \(b = -2\):
- \(x - y - 1 = 2 \Rightarrow x - y = 3\)
- \(2(y - \frac{1}{2}) = -2 \Rightarrow y = 0\)
- \(x = 3\) → \((3, 0)\)

7. For \(a = -2\), \(b = 2\):
- \(x - y - 1 = -2 \Rightarrow x - y = -1\)
- \(2(y - \frac{1}{2}) = 2 \Rightarrow y = 1\)
- \(x = 0\) → \((0, 1)\)

8. For \(a = -2\), \(b = -2\):
- \(x - y - 1 = -2 \Rightarrow x - y = -1\)
- \(2(y - \frac{1}{2}) = -2 \Rightarrow y = 0\)
- \(x = 1\) → \((1, 0)\)

### Step 6: List all integer pairs

The integer pairs \((x, y)\) that satisfy the equation are:

1. \((4, 3)\)
2. \((0, -1)\)
3. \((4, 1)\)
4. \((3, 0)\)
5. \((0, 1)\)
6. \((1, 0)\)

### Conclusion

Thus, the total number of integer pairs \((x, y)\) that satisfy the equation is **6**.

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