A quadratic function is one expression that can answer four different questions. Where does it cross the axis, how many times, where is its turning point, and for which values is it above or below zero.
How can one quadratic tell several stories?
Take this example. Take f(x) = x² + 4x − 5.
Roots. Factorise: (x + 5)(x − 1) = 0, so x = −5 or x = 1. The sum of the roots is −4 and the product is −5, which match the coefficients of the equation x² + 4x − 5 = 0.
Discriminant. b² − 4ac = 16 − 4(1)(−5) = 36, which is positive. So there are two different real roots, as found.
Turning point. Complete the square: f(x) = (x + 2)² − 9. The minimum value is −9, at x = −2. That is halfway between the roots −5 and 1.
Inequality. Because the graph opens upward and crosses the axis at −5 and 1, f(x) > 0 when x < −5 or x > 1.
Four questions were answered from one expression. The skills are connected, and the lessons below practise each in turn.
How do the four lessons connect?
| Lesson | The question it answers | Builds on |
|---|---|---|
| Relating roots to coefficients | What do the roots add up to and multiply to? | Factorising |
| Using the discriminant | How many real roots are there? | Solving quadratics |
| Completing the square | Where is the highest or lowest point? | Expanding brackets |
| Quadratic inequalities | For which x is it above or below zero? | Roots and the graph shape |
Who should start where?
A student who is new to the chapter starts with roots and coefficients. A student who finds inequalities confusing should check that the roots lesson is secure first, since every interval starts from the roots.
If the algebra itself slows you down, use the algebra step repair trainer. To see a graph respond as you change the numbers, try the quadratic graph and roots explorer. When you want mixed questions, use the practice set.
This chapter sits alongside functions in the Form 4 topics, and progressions uses the same careful algebra.
Will the pace suit a student who needs to revisit algebra?
Yes. Each lesson uses small whole numbers first and shows every expansion and rearrangement, so the new idea is not buried under arithmetic. Students who need extra time can repeat the self-check question before moving on.
When is tuition worth considering?
Free lessons explain each skill. A one-to-one teacher sees which question types you avoid and works on those first.
Read about online one-to-one Additional Mathematics tuition or return to the Additional Mathematics guide.