Quadratic Systems — SAT Math Explained
Systems of equations where at least one equation is quadratic (nonlinear). Solutions are the points where the curves intersect, and there may be 0, 1, or 2 solutions.
The Core Idea
Where linear systems always have 0 or 1 or ∞ solutions, quadratic systems can have 0, 1, or 2 solutions depending on how many times the curves intersect.
Solution Types
The curves intersect at two distinct points
The curves are tangent — they just touch at one point
The curves don't intersect at all
Solving Method
1. Use substitution — solve the linear equation for one variable (if one equation is linear)
2. Substitute into the quadratic equation
3. Solve the resulting quadratic (you'll get a quadratic in one variable)
4. Find both x-values
5. Substitute each x back to find corresponding y-values
6. Write each solution as an ordered pair
Graphical Interpretation
The solutions are the coordinates of the intersection points of the two curves
Common Errors to Avoid
Expecting exactly one solution like a linear system
Not finding the y-coordinates for both x-solutions
Forgetting to check solutions in both equations
Practice: Quadratic Systems
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How many solutions can a system with one quadratic and one linear equation have?
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