Discriminant is one of 4 quadratics formulas in the algebra section of this library, and it is used at high school level.
Why discriminant works
This is exactly the quantity sitting under the square root in the quadratic formula, pulled out and named. A positive value has a real square root that gets added and subtracted, producing two answers. Zero adds nothing, so the two answers merge. A negative value has no real square root, so the answers pick up imaginary parts.
What each symbol means
$a,b,c$ come from $ax^2+bx+c=0$.
Discriminant: when it holds
$\Delta>0$ gives two real roots, $\Delta=0$ one repeated root, and $\Delta<0$ two complex roots.
When it stops applying
The two-real, one-real, none-real reading only holds when a, b and c are real numbers. With complex coefficients the sign of this expression means nothing, because complex numbers are not ordered as positive or negative in the first place.
Discriminant: a worked example
For $x^2-6x+9$, $\Delta=36-36=0$, so $x=3$ is repeated.
The mistake to avoid
What people do: Computing b squared as a negative number when b itself is negative, such as -36 for b = -6.
Why it goes wrong: Squaring destroys the sign, so (-6) squared is 36. Carrying a wrong sign here can turn two real roots into a report of no real roots.
Do this instead: Put the coefficient in brackets before squaring it, and remember that only the -4ac part of the expression is ever allowed to be negative.
Discriminant: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b,c$ come from $ax^2+bx+c=0$.
- Check the conditions before substituting. $\Delta>0$ gives two real roots, $\Delta=0$ one repeated root, and $\Delta<0$ two complex roots.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most algebra slips.
Where this formula fits
- Subject
- Algebra formulas — 28 entries in this library
- Topic
- Quadratics
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where discriminant comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- The Quadratic Formula Explained — the lesson behind this formula: use the formula and interpret its discriminant.
- Quadratic Formula Calculator — check your substitution and the value it produces.
- Study algebra — the subject guide that explains the ideas these formulas compress.
- Algebra I Practice — questions that make you retrieve the formula instead of recognising it.
- All 28 algebra formulas — the full grouped reference, or the complete formula library.
Questions about discriminant
Does the discriminant tell me what the roots actually are?
No, only how many there are and what kind. To get the values themselves you still need the full quadratic formula, factoring, or the square-root method.
If the discriminant is 20, are the roots irrational?
Yes, when the coefficients are rational. For x squared minus 6x plus 4 the discriminant is 20, and the square root of 20 is not a whole number, so the roots 3 plus or minus the square root of 5 cannot be written as fractions.
What does the sign mean on the graph?
It counts the crossings of the horizontal axis. Positive means the parabola cuts through twice, zero means it just touches once, and negative means it floats entirely above or entirely below.
Can I use it to check whether an expression will factor?
Yes. With integer coefficients, the quadratic factors over the rationals exactly when the discriminant is a perfect square, so a value like 20 tells you to stop hunting for nice factors.