Cambridge O Level Additional Mathematics · Syllabus 4037 · 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 O Level Additional Mathematics.
Discriminant in context
The discriminant \(b^2-4ac\) of a quadratic equation \(ax^2+bx+c=0\) tells you how many real roots it has without solving it: positive gives two distinct real roots, zero gives two equal roots, and negative gives no real roots. When a line and a curve are combined into a single quadratic, the same three cases mean the line cuts the curve twice, is a tangent to it, or misses it entirely. Always merge the two equations first and take the discriminant of the combined quadratic, never of the curve on its own.
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.
Questions students ask about Discriminant
What does the discriminant tell you?
The discriminant \(b^2-4ac\) tells you how many real roots \(ax^2+bx+c=0\) has without solving it. If \(b^2-4ac>0\) there are two distinct real roots; if it equals \(0\) the roots are equal; if it is negative there are no real roots. For a line meeting a curve, combine them into one quadratic first: the same three cases mean the line cuts the curve twice, touches it as a tangent, or misses it. Say “no real roots”, not “imaginary roots”, in 4037.
When do you use the discriminant to find an unknown constant?
Whenever a question says a line is a tangent to a curve, meets it twice, or does not meet it, or says an equation has equal roots or no real roots. Combine the line and curve into a single quadratic equal to zero, form \(b^2-4ac\) with the coefficients of that combined equation, and apply the stated condition. If the coefficient of \(x^2\) contains the unknown, state that it is non-zero; and when you take a square root, write \(\pm\) so you keep both values.