**Discriminant** | **Real roots** Delta=b2-4ac>0 | Two distinct Delta=0 | One repeated Delta<0 | None Delta>0 Delta=0 Delta<0 First check a 0. If a parameter makes the quadratic coefficient zero, examine that resulting equation separately. Example For x2+kx+9=0, two distinct real roots require k2-36>0, giving k<-6 or k>6. Equal roots occur when k=+-6. For a 0, two distinct roots mean the curve crosses the horizontal axis twice; a repeated root means it touches the axis once; no real roots mean it never meets the axis. If Delta=0, the quadratic is non-negative for all real x when a>0, and non-positive when a<0. If Delta<0, it is strictly positive for all real x when a>0, and strictly negative when a<0.

First check \(a\ne0\). If a parameter makes the quadratic coefficient zero, examine that resulting equation separately.

For \(x^2+kx+9=0\), two distinct real roots require \(k^2-36>0\), giving \(k<-6\) or \(k>6\). Equal roots occur when \(k=\pm6\).

For \(a\ne0\), two distinct roots mean the curve crosses the horizontal axis twice; a repeated root means it touches the axis once; no real roots mean it never meets the axis. If \(\Delta=0\), the quadratic is non-negative for all real \(x\) when \(a>0\), and non-positive when \(a<0\). If \(\Delta<0\), it is strictly positive for all real \(x\) when \(a>0\), and strictly negative when \(a<0\).

Quadratic Functions, Equations and Inequalities

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

The discriminant

The discriminant

Discriminant	Real roots
\(\Delta=b^2-4ac>0\)	Two distinct
\(\Delta=0\)	One repeated
\(\Delta<0\)	None