Sure, let's work through the equation [tex]\( x^2 - 2x - 2y - y^2 = 0 \)[/tex] step by step to find its solutions.
### Step 1: Set Up the Equation
We start with the quadratic equation:
[tex]\[ x^2 - 2x - 2y - y^2 = 0 \][/tex]
### Step 2: Rewriting the Equation
We will rearrange and complete the square for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
#### Completing the Square for [tex]\( x \)[/tex]:
For the [tex]\( x \)[/tex] terms, [tex]\( x^2 - 2x \)[/tex]:
[tex]\[ x^2 - 2x = (x - 1)^2 - 1 \][/tex]
#### Completing the Square for [tex]\( y \)[/tex]:
For the [tex]\( y \)[/tex] terms, [tex]\(-y^2 - 2y \)[/tex]:
[tex]\[ -y^2 - 2y = -(y^2 + 2y + 1 - 1) = -(y + 1)^2 + 1 \][/tex]
### Step 3: Substitution
Substitute the completed square expressions back into the equation:
[tex]\[ (x - 1)^2 - 1 - (y + 1)^2 + 1 = 0 \][/tex]
Simplify:
[tex]\[ (x - 1)^2 - (y + 1)^2 = 0 \][/tex]
### Step 4: Factoring
Notice that this is now a difference of squares:
[tex]\[ (x - 1)^2 = (y + 1)^2 \][/tex]
We can factor it as:
[tex]\[ (x - 1 + (y + 1))(x - 1 - (y + 1)) = 0 \][/tex]
Simplify the factors:
[tex]\[ (x + y)(x - y - 2) = 0 \][/tex]
### Step 5: Solving the Factors
Now, solve for [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in each case.
#### Case 1: [tex]\( x + y = 0 \)[/tex]
[tex]\[ x = -y \][/tex]
#### Case 2: [tex]\( x - y - 2 = 0 \)[/tex]
[tex]\[ x = y + 2 \][/tex]
### Step 6: Express Solutions
We have found two sets of solutions for the equation:
1. [tex]\( x = -y \)[/tex]
2. [tex]\( x = y + 2 \)[/tex]
Thus, the complete set of solutions is:
[tex]\[ (x, y) = (-y, y) \quad \text{and} \quad (x, y) = (y + 2, y) \][/tex]
### Conclusion
The solutions to the equation [tex]\( x^2 - 2x - 2y - y^2 = 0 \)[/tex] are:
[tex]\[ (x, y) = (-y, y) \quad \text{and} \quad (x, y) = (y + 2, y) \][/tex]
These represent the relationships between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] for which the given quadratic equation holds true.
