Discriminant (AA SL)
The discriminant \(\Delta = b^2 - 4ac\) tells you how many real roots a quadratic has before you solve anything. One calculation, one sign check, and you know whether the graph crosses the \(x\)-axis twice, touches it once, or misses it altogether. This page covers the formula, the three root cases, worked examples, and the mistakes that cost marks. It's part of the broader Quadratics & Equations topic.
26 questions on this sub-topic.
The discriminant and the three cases
Covered under IB syllabus reference SL2.7: solution of quadratic equations and inequalities by factorisation, completing the square, and the quadratic formula, together with the discriminant \(\Delta = b^2 - 4ac\) and the nature of the roots.
The discriminant
\(\Delta = b^2 - 4ac\)
For \(ax^2 + bx + c = 0\). This formula is in the booklet, so you don't need to memorise it - just recognise when a question wants it.
Nature of the roots
\(\Delta > 0\): two distinct real roots.
\(\Delta = 0\): one repeated real root.
\(\Delta < 0\): no real roots.
This case split isn't itself a booklet formula - it's the interpretation you're expected to know cold.
Need the full syllabus wording, the quadratic formula, and vertex form? See Quadratics & Equations.
Worked examples
Find the value of \(k\) for which \(x^2+kx+9=0\) has equal roots, given \(k>0.\)
Worked solution
Equal roots means the discriminant is zero: \(\Delta = k^2 - 4(1)(9) = k^2 - 36 = 0.\) M1
Rearranging gives \(k^2 = 36.\) A1
Since \(k > 0\), take the positive root: \(k = 6.\) A1
Find the values of \(k\) for which \(x^2 + kx + 4 = 0\) has two distinct real roots.
Worked solution
Two distinct real roots requires \(\Delta = b^2 - 4ac > 0.\) M1
Substitute \(a=1,\ b=k,\ c=4\): \(k^2 - 16 > 0.\) A1
Solving the inequality: \(k^2 > 16 \Rightarrow k < -4\) or \(k > 4.\) A1
Both branches are required, since either sign of \(k\) can satisfy \(k^2 > 16.\) R1
Common mistakes
- Not rearranging to \(=0\) before reading off \(a\), \(b\), \(c\). If the equation is given as \(x^2+3x=5\), you must rewrite it as \(x^2+3x-5=0\) before substituting into the discriminant.
- Mixing up which sign gives which case. \(\Delta>0\) means two roots exist, not zero - it's easy to read the inequality backwards under pressure. Say the case out loud as you write it: "positive means two roots."
- Treating "distinct" and "real" as the same condition. "Two distinct real roots" needs the strict inequality \(\Delta>0\); "real roots" (allowing a repeated one) only needs \(\Delta\ge0\). Read the question wording carefully before choosing \(>\) or \(\ge\).
Ready to practise properly?
25 discriminant questions, marked instantly like the real exam.
Quick answers
What is the discriminant of a quadratic equation?
For \(ax^2 + bx + c = 0\), the discriminant is \(\Delta = b^2 - 4ac\). Its sign tells you how many real roots the equation has, without needing to solve it.
What does the sign of the discriminant tell you?
\(\Delta > 0\) gives two distinct real roots, \(\Delta = 0\) gives one repeated real root, and \(\Delta < 0\) gives no real roots (the graph doesn't cross the \(x\)-axis). For GDC guidance on solving and graphing quadratics, see the parent topic's using your GDC section.