Identify equation: Identify the equation to be solved. The equation is 0=-(y^(2)-2y+x).Rewrite without negative sign: Rewrite the equation without the negative sign by multiplying both sides by -1 to get a standard quadratic form. The equation becomes y^(2)-2y+x=0.Recognize quadratic form: Recognize that the equation is a quadratic equation in the form of ay^2 + by + c = 0, where a=1, b=-2, and c=x.Use quadratic formula: To solve the quadratic equation, we can either factor it, complete the square, or use the quadratic formula. Since we do not have specific values for y and x, we will use the quadratic formula: y = (-b ± √(b^2 - 4ac)) / (2a).Substitute values into formula: Substitute the values of a, b, and c into the quadratic formula. This gives us y = (-(-2) ± √((-2)^2 - 4*1*x)) / (2*1).Simplify equation: Simplify the equation by performing the operations inside the formula. This gives us y = (2 ± √(4 - 4x)) / 2.Further simplify equation: Further simplify the equation by dividing the terms inside the square root as well as the 2 outside. This gives us y = 1 ± √(1 - x).

Calculus

Find derivatives using the chain rule I

Full solution

0=-\left(y^{2}-2 y+x\right)

Identify equation: Identify the equation to be solved. The equation is $0=-(y^{2}-2y+x)$ .
Rewrite without negative sign: Rewrite the equation without the negative sign by multiplying both sides by $-1$ to get a standard quadratic form. The equation becomes $y^{2}-2y+x=0$ .
Recognize quadratic form: Recognize that the equation is a quadratic equation in the form of $ay^2 + by + c = 0$ , where $a=1$ , $b=-2$ , and $c=x$ .
Use quadratic formula: To solve the quadratic equation, we can either factor it, complete the square, or use the quadratic formula. Since we do not have specific values for $y$ and $x$ , we will use the quadratic formula: $y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Substitute values into formula: Substitute the values of $a$ , $b$ , and $c$ into the quadratic formula. This gives us $y = \frac{-(-2) \pm \sqrt{(-2)^2 - 4\cdot1\cdot x}}{2\cdot1}.$
Simplify equation: Simplify the equation by performing the operations inside the formula. This gives us $y = \frac{2 \pm \sqrt{4 - 4x}}{2}$ .
Further simplify equation: Further simplify the equation by dividing the terms inside the square root as well as the $2$ outside. This gives us $y = 1 \pm \sqrt{1 - x}$ .