The equation of an intersection

At an intersection, both graphs have the same x and y. Substitute the line equation into the curve, then use the resulting equation to count or find common points. For a full quadratic curve, a repeated root of the intersection equation means the line touches the curve. Two distinct roots mean two common points. No real root means no common point. Equate the expressions for y, rearrange to zero, calculate the discriminant or solve, then substitute each accepted x into the line. Finally check the original domain. Example The line y=2x-1 and curve y=x2 give (x-1)2=0. The contact point is (1,1). Exam check: D belongs to the intersection equation, not the curve alone. A restricted curve can retain only one of two intersection roots without being tangent.