Discriminant and x-axis intersections

The number of points a parabola shares with the xx-axis is decided by the sign of the discriminant D=b2−4acD = b^2 - 4ac1.

Why DD alone decides it

The shared points have xx-coordinates solving ax2+bx+c=0ax^2 + bx + c = 0, and the quadratic formula reads as follows.

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the radical is exactly DD.

DDD\sqrt{D}Solutions
D>0D > 0a positive real numbertwo, split by the ±\pm
D=0D = 000one repeated root −b2a-\dfrac{b}{2a}
D<0D < 0not realnone

Three parabolas compared

ParabolaDDShared points
y=x2−2x−3y = x^2 - 2x - 31616x=−1x = -1 and x=3x = 3
y=x2−2x+1y = x^2 - 2x + 100tangent at (1,0)(1, 0)
y=x2−2x+2y = x^2 - 2x + 2−4-4none

The same parabola at three heights

The three are in fact one parabola raised and lowered. Completing the square turns them into (x−1)2−4(x - 1)^2 - 4, (x−1)2(x - 1)^2 and (x−1)2+1(x - 1)^2 + 1: every vertex sits above x=1x = 1, and only the height changes.

Vertex heightCrossings
−4-4 (below the axis)two
00 (on the axis)tangency
11 (above the axis)none

What the discriminant really reports is the height of the vertex. Indeed the vertex has yy-coordinate −D4a-\dfrac{D}{4a}, which here comes to −4-4, 00 and 11, precisely the three heights.

Beyond the xx-axis

The discriminant is not limited to the xx-axis. Combining a parabola with any line and simplifying gives a quadratic equation, and the sign of its DD decides in the same way whether they cross twice, touch, or miss. That is why D=0D = 0 is the condition for a tangent line.

The large dots mark the shared points.

  1. Discriminant, Wikipedia