Using the Discriminant to Find the Nature of Roots

The discriminant tells you how many real roots a quadratic equation has, and what kind, without actually solving it. For a quadratic written as $ax^2+bx+c=0$, the discriminant is the expression $b^2-4ac$, the part that sits under the square root in the quadratic formula.

Because a real square root only exists for a value that is zero or positive, the sign of $b^2-4ac$ decides everything: positive gives two distinct real roots, zero gives two equal real roots (a repeated root), and negative gives no real roots at all. This method answers questions such as "state the nature of the roots", "find the value of $k$ for which the equation has equal roots", or "show that the roots are always real".

It is a core Form 4 Quadratic Functions skill, closely linked to whether a graph meets the $x$-axis.

Because the discriminant works from the coefficients alone, it is fast and reliable. You never have to factorise or reach for the full formula just to classify the roots, reading the sign of one expression is enough.

Reach for the discriminant whenever a question asks about the number or type of roots rather than their exact values. Phrases such as "nature of the roots", "has two distinct roots", "has equal roots", "has no real roots", or "the roots are real and different" are the direct cue.

It is also the method for questions that contain an unknown, such as "find the range of values of $k$ for which the equation has real roots", because the condition becomes an inequality or equation in $k$. Geometrically, use it when a question asks whether a curve cuts, touches, or misses the $x$-axis, or whether a line is a tangent to a curve.

Nature-of-roots work appears in both Paper 1 and Paper 2, so recognising these cues quickly is worthwhile.

Work through the classification like this:

1. 1

   Write the equation as $ax^2+bx+c=0$

   Move every term to one side so the right-hand side is zero, then read off $a$, $b$ and $c$.

2. 2

   Identify $a$, $b$ and $c$ with their signs

   Be careful with negatives; the coefficient $b$ includes its sign, and so do $a$ and $c$.

3. 3

   Compute the discriminant $b^2-4ac$

   Substitute the coefficients and simplify to a single number or an expression in the unknown.

4. 4

   Compare with zero

   Decide whether $b^2-4ac$ is positive, zero, or negative to classify the roots.

5. 5

   State the conclusion (or solve for the unknown)

   Report the nature of the roots, or set up the matching condition $b^2-4ac>0$, $=0$, or $<0$ and solve for the unknown.

The three cases are summarised here:

Try this yourself first, then check each line against the solution.

Notice that squaring produces two answers, $k=6$ and $k=-6$, and both are valid. A common slip is to keep only the positive value; the check by factorising confirms that each choice really does give a perfect square, so neither can be dropped.

• Not rearranging to $ax^2+bx+c=0$ first. If the equation is not equal to zero, the coefficients $a$, $b$, $c$ will be wrong.
• Getting the sign of $b$ or $c$ wrong. A negative coefficient stays negative inside $b^2-4ac$.
• Reading "equal roots" as "no roots". Equal roots means $b^2-4ac=0$; no real roots means $b^2-4ac<0$.
• Keeping only one value of $k$ after $k^2=36$. Squaring gives both $k=6$ and $k=-6$.
• Flipping the inequality when solving for the unknown, for example writing $b^2-4ac>0$ as the condition for no real roots instead of two distinct roots.

The step that quietly loses marks is the translation, turning the words "equal roots" or "no real roots" into the correct condition on $b^2-4ac$. In a one-to-one lesson our teacher checks that translation with you before any algebra, so you never solve the wrong inequality.

Because our teachers are experienced, you get someone who links the sign of the discriminant to the picture of the parabola cutting, touching, or missing the $x$-axis, so the rule sticks. Lessons are online and taught in English, while SPM papers are set in both Malay and English.

We also drill the habit of keeping both values from a square root, and of checking a chosen coefficient by factorising, so the final line is always secure.