Alright, let's solve the given equation step-by-step:
We start with the equation:
[tex]\[ x^2 + 2y - y^2 - 2x = 0 \][/tex]
We want to solve this equation for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex].
1. Transpose Terms: Rearrange the equation to separate the terms involving [tex]\( y \)[/tex]:
[tex]\[ x^2 - 2x + 2y - y^2 = 0 \][/tex]
We can rewrite it as:
[tex]\[ -y^2 + 2y + x^2 - 2x = 0 \][/tex]
2. Convert to Standard Quadratic Form: Recognize that this is a quadratic equation in terms of [tex]\( y \)[/tex]. Rewrite it in standard quadratic form [tex]\( ay^2 + by + c = 0 \)[/tex] where [tex]\( a = -1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = x^2 - 2x \)[/tex]:
[tex]\[ -y^2 + 2y + (x^2 - 2x) = 0 \][/tex]
3. Solve the Quadratic Equation: To solve for [tex]\( y \)[/tex], we use the quadratic formula, [tex]\( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
Plug in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ y = \frac{-2 \pm \sqrt{(2)^2 - 4(-1)(x^2 - 2x)}}{2(-1)} \][/tex]
Simplify within the square root:
[tex]\[ y = \frac{-2 \pm \sqrt{4 + 4(x^2 - 2x)}}{-2} \][/tex]
[tex]\[ y = \frac{-2 \pm \sqrt{4 + 4x^2 - 8x}}{-2} \][/tex]
[tex]\[ y = \frac{-2 \pm \sqrt{4(x^2 - 2x + 1)}}{-2} \][/tex]
[tex]\[ y = \frac{-2 \pm \sqrt{4(x - 1)^2}}{-2} \][/tex]
[tex]\[ y = \frac{-2 \pm 2(x - 1)}{-2} \][/tex]
4. Simplify the Expressions: Solve for the two possible values of [tex]\( y \)[/tex]:
[tex]\[ y = \frac{-2 + 2(x - 1)}{-2} \quad \text{and} \quad y = \frac{-2 - 2(x - 1)}{-2} \][/tex]
Simplify each case:
[tex]\[ y = \frac{-2 + 2x - 2}{-2} = \frac{2(1 - x)}{-2} = x \][/tex]
[tex]\[ y = \frac{-2 - 2x + 2}{-2} = \frac{2(x - 1)}{-2} = 2 - x \][/tex]
Thus, the solutions for [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex] are:
[tex]\[ y = x \][/tex]
[tex]\[ y = 2 - x \][/tex]
These are the two solutions to the equation [tex]\( x^2 + 2y - y^2 - 2x = 0 \)[/tex].
