This is a system of equations that needs to be solved. The system is:

1. $y = x^2 - x + 2$
2. $y = 0.5x^2 + 3.5$

To find the solution(s), we need to find the values of $x$ and $y$ where both equations are true. Let me solve it. sympy as sp

Define variables

x, y = sp.symbols('x y')

Define the equations

eq1 = sp.Eq(y, x2 - x + 2) eq2 = sp.Eq(y, 0.5*x2 + 3.5)

Solve the system of equations

solution = sp.solve([eq1, eq2], (x, y)) solutionThe solutions to the system of equations are:

1. $x = -1$, $y = 4$
2. $x = 3$, $y = 8$

Would you like me to explain the steps in detail or verify these solutions graphically?

Expanding your knowledge:

1. How would you interpret the solutions graphically?
2. What does the intersection of these two equations represent?
3. How would the solution change if one of the equations was linear?
4. Can such systems have more than two solutions? Why or why not?
5. How does the discriminant of a quadratic equation help predict the number of intersections?

Tip:

When solving equations, substituting one equation into another is a common and effective method!