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.

下一节推荐课程

通过 Systems of Equations in Two Variables 继续学习此主题。

Systems of Equations in Two Variables