Advanced Math / Systems Equations Two Variables
Nonlinear Systems and Intersection Counts
Systems involving a line and a parabola can have zero, one, or two real solutions. Setting the equations equal usually creates a quadratic equation, and the discriminant determines the number of intersections.
Least you need to know
- A line and a parabola intersect where their equations have the same $y$-value.
- After substitution, a positive discriminant means two real solutions.
- A zero discriminant means exactly one real solution.
- A negative discriminant means no real solutions.
Key notation
- $x^2+bx+c=0$ — A quadratic equation whose discriminant is $b^2-4ac$
- $y=x^2+1$ and $y=2x+k$ — A nonlinear system whose solution count depends on $k$
Worked example
- Set $x^2+4=2x+k$ to compare a parabola and a line.
- Rearrange to $x^2-2x+(4-k)=0$.
- For exactly one solution, set the discriminant equal to 0.
Common mistakes
- The number of real solutions to the resulting quadratic equals the number of intersection points.
- Tangency corresponds to exactly one real solution.
- Be careful with signs when moving all terms to one side.
How to recognize it
- The system includes one linear and one quadratic equation.
- The problem asks how many ordered pairs satisfy the system.
- The problem asks for a parameter that gives exactly one or no real solution.
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