Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.x^2 - y = 0 x + y = 2

Verified step by step guidance

1. Step 1: Start by solving one of the equations for one variable. From the second equation, \( x + y = 2 \), solve for \( y \) to get \( y = 2 - x \).
2. Step 2: Substitute the expression for \( y \) from Step 1 into the first equation \( x^2 - y = 0 \). This gives \( x^2 - (2 - x) = 0 \).
3. Step 3: Simplify the equation from Step 2. Distribute and combine like terms to get \( x^2 + x - 2 = 0 \).
4. Step 4: Solve the quadratic equation \( x^2 + x - 2 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1 \), \( b = 1 \), and \( c = -2 \).
5. Step 5: Once you find the values of \( x \), substitute them back into \( y = 2 - x \) to find the corresponding \( y \) values. This will give you the solutions to the system of equations.

Key Concepts

Nonlinear Equations

Nonlinear equations are equations in which the variables are raised to a power greater than one or involve products of variables. Unlike linear equations, which graph as straight lines, nonlinear equations can produce curves, circles, parabolas, and other complex shapes. Understanding how to manipulate and solve these equations is crucial for finding their intersections or solutions.

Systems of Equations

A system of equations consists of two or more equations that share common variables. The goal is to find values for these variables that satisfy all equations simultaneously. Systems can be linear or nonlinear, and solving them often involves substitution or elimination methods, as well as graphical interpretations to identify points of intersection.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i'. In the context of solving equations, complex solutions arise when the discriminant of a quadratic equation is negative, indicating that the solutions cannot be represented on the real number line. Understanding complex numbers is essential for fully addressing all potential solutions in nonlinear systems.