Cambridge IGCSE Additional Mathematics · Syllabus 0606 · Quadratic Functions
Discriminant
What is Discriminant?
The quantity b^2 - 4ac formed from the coefficients of a quadratic equation ax^2 + bx + c = 0, usually written as the Greek capital delta. Its sign alone determines the number of real roots without the equation being solved: positive gives two distinct real roots, zero gives two equal real roots, and negative gives no real roots. Applied to the single quadratic obtained by equating a line and a curve, the same three cases correspond to the line cutting the curve twice, being tangent to it, and not meeting it at all.
This definition is part of the Quadratic Functions chapter in Cambridge IGCSE Additional Mathematics.
Common mistakes with Discriminant
- “For \(y=x^2-4x+7\) and \(y=mx+1\), \(\Delta=(-4)^2-4(1)(7)\).” Why it fails Those are the coefficients of the curve, not of the combined equation. The discriminant is only defined once the two have been merged into \(x^2-(m+4)x+6=0\). Test Ask: which single equation, equal to zero, am I taking the discriminant of? If you cannot write it down, you are not ready to use \(\Delta\).
- “\(\Delta>0\), so the roots are equal.” Why it fails Equal roots need \(\Delta=0\) exactly. A positive discriminant gives two different roots. Test \(\pm\sqrt\Delta\) gives two different numbers unless \(\Delta\) is zero.
- “\(kx^2+4x+k=0\) has equal roots, so \(k=\pm2\) or \(k=0\).” Why it fails \(k=0\) destroys the \(x^2\) term, so the equation is no longer quadratic and has no discriminant. Test Whenever the coefficient of \(x^2\) contains the unknown, state the condition that it is non-zero.