Cuts, touches or misses: lines and quadratic curves

Picture the curve $y = x^2 - 3x + 6$ and a family of parallel lines $y = x + k$. Changing $k$ slides the line up and down like a ruler.

• High up, the line slices through the curve: two meeting points.
• Lower down, it just kisses the curve: one point. That's a tangent.
• Lower still, it sails underneath: none.

How do we count the meeting points with algebra? A meeting point lies on both graphs, so at that point the two $y$-values are equal. Setting the two right-hand sides equal, $x^2 - 3x + 6 = x + k$, gives an equation whose solutions are exactly the $x$-coordinates of the meeting points. Rearranged, it's a quadratic $ax^2 + bx + c = 0$.

Reminder: the discriminant. The quadratic formula is

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.$

The part under the square root, $b^2 - 4ac$, is the discriminant. It decides how many solutions there are:

• positive: $\pm$ gives two different answers, so two roots;
• zero: $\pm 0$ gives the same answer twice, so one (repeated) root;
• negative: a negative number has no real square root, so no real roots.

So: number of meeting points = number of real roots of that one quadratic. The discriminant decides.

The discriminant belongs to the combined quadratic (after you set the equations equal and rearrange to $= 0$), not to the curve on its own.