Systems of a Linear and a Nonlinear Equation

A line is degree 1, a parabola or circle is degree 2. When you combine them, the highest power that survives is x², so the system boils down to one quadratic equation. A quadratic has at most two roots, which is why a line usually crosses a parabola at two points, touches it at one, or misses it completely. Substitution is the tool: solve one equation for y (or use it as is), set the two expressions for y equal, and move everything to one side.

Once you have the quadratic ax² + bx + c = 0, the discriminant b² - 4ac tells you how many real intersection points exist. Positive means two crossing points, zero means the line is tangent (one point), negative means no real intersection at all. This is exactly how you find a value like c that makes a line touch a parabola exactly once: build the quadratic, set the discriminant to zero, and solve for the unknown.

A circle changes the picture only in shape, not in method. Substituting a line into x² + y² = r still gives a single quadratic in x, so the same three outcomes apply: two solutions if the line cuts through, one if it grazes the edge, none if it passes outside. This topic sits next to pure quadratic solving and the discriminant work from that unit, and it previews how later problems ask you to reason about a system geometrically before grinding through algebra.