A line and a parabola can meet twice, once, or not at all. Drag the line's slope and intercept to see how the number of intersection points changes — and how that connects to solving the system algebraically.
Solving a quadratic equation means finding where it equals zero — its roots. The fastest method, when it works, is factoring: rewrite the expression as a product of two binomials, then set each one to zero. If x² + 2x − 15 factors as (x + 5)(x − 3), the equation is zero whenever either factor is zero.
A linear-quadratic system asks where a line and a parabola intersect. Substituting the line’s equation into the quadratic gives a single quadratic equation to solve — its solutions are the x-coordinates of the intersection points. The discriminant (the part under the square root in the quadratic formula) tells you how many real solutions to expect before you even finish solving: positive means two, zero means exactly one (the line is tangent to the curve), and negative means none.