Consider the following two quadratic equations in the variables x and y:
Equation 1: \(x^2 + y^2 – 4x + 2y – 1 = 0\)
Equation 2: \(2x^2 – y^2 + 3x – 5y + 2 = 0\)
(a) Form a system of equations using the equations provided.
(b) Solve the system simultaneously using appropriate methods.
(c) Discuss the possibility of obtaining complex solutions from the system if real intersections do not exist.
Question
Form a system of equations by clearly writing down and simplifying the given quadratic equations: \(x^2 + y^2 – 4x + 2y – 1 = 0\) and \(2x^2 – y^2 + 3x – 5y + 2 = 0\).
Hint
Hint
Hint
Start by writing each equation as provided and ensure all terms are correctly grouped.
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Complete Answer
The system is:
1) \(x^2 + y^2 – 4x + 2y – 1 = 0\)
2) \(2x^2 – y^2 + 3x – 5y + 2 = 0\)
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Question
Solve the system simultaneously using an appropriate method (such as elimination or substitution). Provide approximate solutions to two decimal places.
Hint
Hint
Hint
Consider combining the equations to eliminate one variable or isolating one variable in one equation and substituting into the other.
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Complete Answer
The approximate real solutions are \( (x,y) \approx (1.25, 1.49) \) and \( (x,y) \approx (-0.14, 0.40) \).
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Question
Discuss the possibility of obtaining complex solutions from the system if no real intersections were found. Explain graphically what complex solutions represent in the context of these quadratic curves.
Hint
Hint
Hint
Examine the discriminant of the derived quadratic equations from the elimination process, or consider the nature of roots of polynomial equations and their graphical interpretation.
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If the discriminant of the quartic equation (or effectively a quadratic equation after elimination of one variable leading to a higher degree polynomial) obtained during the solution process indicates no real roots, then the solutions for \(x\) (and consequently for \(y\)) would be complex. Graphically, real solutions represent intersection points of the curves in the real \(xy\)-plane. Complex solutions, in contrast, imply that the curves do not intersect in the real plane, but would intersect in the complex plane. These complex intersections are not visualizable on a standard real Cartesian plane.
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