Advanced Math / Systems Nonlinear Equations
Line-parabola intersections
A line-parabola system can be solved by setting the two expressions for $y$ equal. The resulting quadratic tells how many intersection points exist.
Least you need to know
- Two real roots give two intersections.
- One repeated root gives one intersection.
- No real roots give no real intersection points.
Key notation
- D — The discriminant $b^2-4ac$ of a quadratic
Worked example
If $x^2=2x+8$, then $x^2-2x-8=0$, or $(x-4)(x+2)=0$. The graphs intersect at two $x$-values.
Common mistakes
- A repeated root is one solution, not two distinct points.
- A line and a parabola cannot intersect in more than two points.
How to recognize it
- A system includes one linear equation and one quadratic equation.
- The answer choices are numbers of solutions or parameter conditions.
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