To solve the equation [tex]\( x^2 + y^2 + 2x - 2y - 7 = 0 \)[/tex], we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy this equation. Here is a step-by-step breakdown:
1. Rewrite the equation: The given equation is:
[tex]\[
x^2 + y^2 + 2x - 2y - 7 = 0
\][/tex]
2. Complete the square: To simplify, we can complete the square for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
For [tex]\( x \)[/tex]:
[tex]\[
x^2 + 2x = (x + 1)^2 - 1
\][/tex]
For [tex]\( y \)[/tex]:
[tex]\[
y^2 - 2y = (y - 1)^2 - 1
\][/tex]
Substitute these into the equation:
[tex]\[
(x + 1)^2 - 1 + (y - 1)^2 - 1 - 7 = 0
\][/tex]
3. Combine constants: Simplify the equation by combining the constants:
[tex]\[
(x + 1)^2 + (y - 1)^2 - 9 = 0
\][/tex]
[tex]\[
(x + 1)^2 + (y - 1)^2 = 9
\][/tex]
4. Analyze the equation: This is the equation of a circle centered at [tex]\((-1, 1)\)[/tex] with radius 3. To find the points where this circle intersects with possible [tex]\(x\)[/tex] and [tex]\( y \)[/tex]:
5. Express [tex]\(x\)[/tex] in terms of [tex]\(y\)[/tex]: Solve for [tex]\(x\)[/tex] explicitly by rearranging the equation:
[tex]\[
(x + 1)^2 = 9 - (y - 1)^2
\][/tex]
Take the square root on both sides:
[tex]\[
x + 1 = \pm \sqrt{9 - (y - 1)^2}
\][/tex]
Thus, we get two solutions for [tex]\( x \)[/tex]:
[tex]\[
x = -1 \pm \sqrt{9 - (y - 1)^2}
\][/tex]
6. Substitute [tex]\( y \)[/tex] back: To understand the explicit solutions, let's rewrite it in a clearer form:
[tex]\[
x = -1 + \sqrt{9 - (y - 1)^2}, \quad x = -1 - \sqrt{9 - (y - 1)^2}
\][/tex]
7. Implicit condition on [tex]\(y\)[/tex]: To ensure real solutions, the expression under the square root must be non-negative:
[tex]\[
9 - (y - 1)^2 \geq 0
\][/tex]
[tex]\[
(y - 1)^2 \leq 9
\][/tex]
[tex]\[
-3 \leq y - 1 \leq 3
\][/tex]
[tex]\[
-2 \leq y \leq 4
\][/tex]
Thus, in conclusion, the solutions to the equation [tex]\( x^2 + y^2 + 2x - 2y - 7 = 0 \)[/tex] are of the form:
[tex]\[
(-1 + \sqrt{9 - (y-1)^2}, y) \quad \text{and} \quad (-1 - \sqrt{9 - (y-1)^2}, y) \quad \text{for} \quad y \in [-2, 4]
\][/tex]
In simplified form:
[tex]\[
(-\sqrt{-(y - 4)(y + 2)} - 1, y) \quad \text{and} \quad (\sqrt{-(y - 4)(y + 2)} - 1, y)
\][/tex]
