Number of Solutions in Nonlinear Systems

Possible Numbers of Solutions

For a linear-quadratic system (line and parabola): 0, 1, or 2 solutions.

For two quadratic equations (two parabolas): 0, 1, 2, 3, or 4 solutions (though SAT typically focuses on 0, 1, or 2).

Method: Set Equal and Use the Discriminant

1. Set the two equations equal (eliminate $y$).
2. Rearrange to standard quadratic form: $ax^2 + bx + c = 0$.
3. Compute $\Delta = b^2 - 4ac$.
4. $\Delta > 0$: 2 solutions. $\Delta = 0$: 1 solution. $\Delta < 0$: 0 solutions.

Graphical Reasoning

You can also reason about intersection counts from graphs:

• A horizontal line $y = k$ intersects a parabola $y = x^2$ at 2 points if $k > 0$, at 1 point if $k = 0$, and at 0 points if $k < 0$.
• A steeper line might intersect a parabola at different points than a shallower line.

Parameter Problems

"For what value of $k$ does the system have exactly one solution?" → Set $\Delta = 0$ and solve for $k$.

"For what values of $k$ does the system have no solution?" → Set $\Delta < 0$ and solve the inequality.

Tip 1: You Don't Need to Solve — Just Count

The discriminant tells you the count without solving.

Tip 2: Sketch a Quick Graph

Even a rough sketch helps you visualise: does the line intersect the parabola?

Tip 3: Use Desmos to Verify

Graph both equations on the calculator to see intersections directly.