SPM Mathematics: How to Master Quadratic Functions and Equations

Quadratics are solved three ways, factorising, the formula, and completing the square, and each is best for a different job. Factorising is fastest when it works; the formula always works; completing the square reveals the turning point for maximum and minimum problems.

Knowing which to reach for is half the skill.

A quadratic can hide inside a KBAT problem about maximum area, inside a graph question, or inside a real-situation word problem. That is why this chapter repays solid work: it is not a topic you finish and forget, but a tool you keep using.

Quadratic inequality questions ask for the range of x where a quadratic expression is positive, negative, or bounded by a specific value, and the fastest reliable route is always through the sketch, not through algebra alone. Once you've found where the parabola crosses the x-axis, the same roots you'd get from factorising or the quadratic formula, mark those two points on a rough sketch and note whether the curve opens upward or downward from the sign of a.

Reading the inequality off that sketch is then almost mechanical: for an upward-opening parabola, the expression is negative between the two roots and positive outside them; for a downward-opening one, it's the reverse. The two most common ways this goes wrong are forgetting that the direction flips when a is negative, and writing the roots in the wrong order when stating the range (the smaller root always comes first in a "between" inequality).

A rough sketch costs seconds and removes both of these errors almost entirely, which is exactly why relying on the picture beats trying to reason about signs from the equation alone.

A quadratic graph question rewards you for reading the equation, not for plotting ten points and hoping. From the completed-square form, the turning point and whether the parabola opens upward or downward come straight from the sign of a.

From the factorised form, the x-intercepts are immediate. The y-intercept is always just c, read off the general form.

Put those three facts together, turning point, direction, and where it crosses each axis, and you can sketch a reasonably accurate curve in under a minute, which is exactly what Paper 1's speed and Paper 2's diagram-labelling both need. The habit to build is checking your sketch against all three forms rather than trusting just one: if your factorised roots don't line up with the axis of symmetry from your completed-square turning point, one of your two workings has an error, and it is much cheaper to catch it here than after the exam.

The discriminant, b² − 4ac, tells you how many real roots a quadratic equation has without solving it, positive means two distinct roots, zero means one repeated root, negative means no real roots. It sounds like a small fact, but it is the tool behind a specific, recurring SPM question type: "find the range of values of k such that the equation has two distinct roots" or "show that the equation has no real roots for all values of k."

These questions are really just inequality problems wearing a quadratic disguise. Set up the discriminant condition, substitute in the given expression, and solve the resulting inequality in k, the quadratic content is already done by that point.

Students often lose marks here not from the algebra but from misreading which condition the question wants: "distinct roots" needs strictly greater than zero, "equal roots" needs exactly zero, and mixing those two up is one of the more common, entirely avoidable slips in this part of the chapter.