Quadratic Functions and Equations in One Variable · Form 4
Roots and the discriminant, explained
The roots of a quadratic equation are the x-values that make ax² + bx + c = 0, and the discriminant b² − 4ac tells you how many of those roots exist before you even solve it.
Roots are where the value is zero
The roots of a quadratic equation are the values of x that make ax² + bx + c equal to zero. On the graph they are exactly the points where the parabola crosses the x-axis, because that axis is the line f(x) = 0.
A quadratic can cross the axis twice, touch it once, or miss it completely, so it may have two roots, one repeated root, or no real roots.
The discriminant counts the roots for you
The discriminant is the part under the square-root sign in the quadratic formula: b² − 4ac. Its sign settles how many real roots there are without solving the whole equation.
If b² − 4ac > 0 there are two different roots; if it equals 0 there is one repeated root and the parabola just touches the x-axis; if it is negative there are no real roots and the curve never reaches the axis. For x² − 4x + 4, for instance, b² − 4ac = 16 − 16 = 0, so there is one repeated root.
Recognising a discriminant question
When a question says the equation has two equal roots, has no real roots, or the curve does not intersect the x-axis, it is really asking you to think about b² − 4ac. A frequent misunderstanding is to read no real roots as no graph, the parabola is still there, it simply floats clear of the x-axis.
Another is to treat one repeated root as no root at all; the root exists, the two solutions just land on the same point.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
What does the discriminant actually predict?
The discriminant, b² − 4ac, tells you the type of roots before solving: positive gives two different real roots, zero gives one repeated (equal) root, and negative gives no real roots. Paper 1 often asks you to judge this straight from the value, without solving the equation at all.
What's the common pitfall when computing the discriminant?
Sign errors from a negative b or negative c are the most common mistake, always substitute values in brackets, e.g. b² − 4ac with b = −3 becomes (−3)² not −3².
Forgetting to rearrange the equation into ax² + bx + c = 0 first also gives the wrong a, b, c.
How does the discriminant connect to the graph without drawing it?
The discriminant tells you how many times the parabola crosses the x-axis: two crossings when positive, one touching point when zero, and no crossing at all when negative. This lets you answer "does the graph cut the x-axis" questions purely by calculation, saving time in the exam.