🔢 Discriminant Calculator
By ToolNimba Math Team · Updated 2026-06-19
Find the discriminant of ax2 + bx + c = 0. Enter the coefficients below.
Enter coefficients a, b and c to find the discriminant.
The discriminant is the part of the quadratic formula under the square root sign: D = b^2 - 4ac. It tells you, before you solve anything, how many real roots the quadratic ax^2 + bx + c = 0 has and whether they are real or complex. Enter the coefficients a, b and c, and this calculator returns the discriminant along with a plain reading of what it means for the roots.
What is the Discriminant Calculator?
Every quadratic equation can be written in the standard form ax^2 + bx + c = 0, where a is not zero. The discriminant, written D or sometimes the Greek delta, is defined as D = b^2 - 4ac. It is exactly the expression that sits beneath the square root in the quadratic formula x = (-b plus or minus the square root of (b^2 - 4ac)) ÷ 2a. Because the square root governs the whole behaviour of the formula, the sign of D decides the entire character of the solution without you having to finish the calculation.
There are three cases. When D is greater than 0 the square root is a real positive number, so the plus and minus branches give two different real roots, and the parabola crosses the x-axis twice. When D equals 0 the square root is zero, the plus and minus branches collapse into one value, and you get a single repeated real root (a double root) where the parabola just touches the x-axis. When D is less than 0 the square root is of a negative number, which has no real value, so the two roots are complex conjugates and the parabola never meets the x-axis.
There is also a useful fourth observation for equations with whole-number coefficients: if D is a positive perfect square (such as 1, 4, 9, 16) then the two real roots are rational and the quadratic factors neatly over the integers. If D is positive but not a perfect square, the roots are irrational. This is why the discriminant is a quick first test in algebra: one subtraction tells you how many solutions exist, whether they are real or complex, and whether the expression will factor cleanly.
When to use it
- Checking how many real solutions a quadratic equation has before solving it in full.
- Deciding whether a quadratic will factor neatly, by testing if the discriminant is a perfect square.
- Finding the value of an unknown coefficient that makes an equation have exactly one (repeated) root, by setting D = 0.
- Verifying homework or exam answers about the nature of the roots of ax^2 + bx + c = 0.
How to use the Discriminant Calculator
- Enter coefficient a, the number in front of x^2 (it must not be zero for a quadratic).
- Enter coefficient b, the number in front of x.
- Enter coefficient c, the constant term.
- Read off the discriminant D, the number of real roots, and the nature of the roots, which update instantly.
Formula & method
Worked examples
Find the discriminant of x^2 - 5x + 6 = 0 (a = 1, b = -5, c = 6).
- D = b^2 - 4ac
- b^2 = (-5)^2 = 25
- 4ac = 4 x 1 x 6 = 24
- D = 25 - 24 = 1
Result: D = 1, which is positive and a perfect square, so there are two distinct rational real roots.
Find the discriminant of x^2 - 4x + 4 = 0 (a = 1, b = -4, c = 4).
- D = b^2 - 4ac
- b^2 = (-4)^2 = 16
- 4ac = 4 x 1 x 4 = 16
- D = 16 - 16 = 0
Result: D = 0, so there is exactly one repeated real root (a double root) at x = 2.
Find the discriminant of x^2 + 2x + 5 = 0 (a = 1, b = 2, c = 5).
- D = b^2 - 4ac
- b^2 = (2)^2 = 4
- 4ac = 4 x 1 x 5 = 20
- D = 4 - 20 = -16
Result: D = -16, which is negative, so the two roots are complex conjugates and the parabola never crosses the x-axis.
What the sign of the discriminant tells you about the roots
| Discriminant D | Number of real roots | Nature of roots | Graph (parabola) |
|---|---|---|---|
| D is greater than 0 | 2 | Two distinct real roots | Crosses the x-axis at two points |
| D = 0 | 1 | One repeated real root (double root) | Touches the x-axis at one point |
| D is less than 0 | 0 | Two complex conjugate roots | Does not meet the x-axis |
Extra detail when coefficients are whole numbers and D is greater than 0
| Condition on D | Roots are | Factors over integers? |
|---|---|---|
| D is a perfect square (1, 4, 9, ...) | Rational | Yes |
| D is positive but not a perfect square | Irrational | No |
Common mistakes to avoid
- Forgetting the sign of b before squaring. b^2 is always positive, even when b is negative. For b = -5, b^2 = 25, not -25. Square the whole coefficient including its sign, then the result is non-negative.
- Mixing up the order in b^2 - 4ac. The discriminant is b^2 minus 4ac, not 4ac minus b^2. Reversing it flips the sign of D and gives the wrong conclusion about the roots.
- Trying to use the discriminant when a = 0. If a = 0 the equation is not quadratic, it is linear (bx + c = 0), and the discriminant b^2 - 4ac does not describe its roots. Make sure a is non-zero first.
- Assuming D = 0 means no solution. A discriminant of zero does not mean there is no root. It means there is exactly one real root, repeated twice (a double root), where the parabola touches the x-axis.
Glossary
- Discriminant
- The expression D = b^2 - 4ac that determines the number and type of roots of a quadratic equation.
- Quadratic equation
- An equation of the form ax^2 + bx + c = 0 where a is not zero.
- Root
- A value of x that makes the equation equal zero, also called a solution or zero of the quadratic.
- Double root
- A single root that is counted twice, occurring when the discriminant is exactly zero.
- Complex conjugate roots
- A pair of roots of the form p + qi and p - qi that appear when the discriminant is negative.
- Perfect square
- A whole number that is the square of an integer, such as 1, 4, 9 or 16. A perfect-square discriminant signals rational roots.
Frequently asked questions
What is the discriminant of a quadratic?
The discriminant is the quantity D = b^2 - 4ac for a quadratic ax^2 + bx + c = 0. It is the part under the square root in the quadratic formula, and its sign tells you how many real roots the equation has and whether those roots are real or complex.
How do you calculate b squared minus 4ac?
Take coefficient b and square it (including its sign, so a negative b becomes positive). Then multiply 4 by a by c. Subtract the second result from the first: D = b^2 - 4ac. This calculator does all three steps for you when you enter a, b and c.
What does a positive discriminant mean?
A positive discriminant (D greater than 0) means the quadratic has two distinct real roots, and its parabola crosses the x-axis at two separate points. If D is also a perfect square and the coefficients are whole numbers, those roots are rational and the expression factors over the integers.
What does it mean when the discriminant is zero?
When D = 0 the quadratic has exactly one real root, repeated twice. This is called a double root or repeated root, and the parabola just touches the x-axis at a single point instead of crossing it.
What if the discriminant is negative?
A negative discriminant (D less than 0) means there are no real roots. The two roots are complex conjugates with an imaginary part, and the parabola never meets the x-axis. The equation still has solutions, but only in the complex numbers.
Can the discriminant be used if a is zero?
No. If a = 0 the equation is not quadratic, it reduces to the linear equation bx + c = 0, and the discriminant b^2 - 4ac no longer describes its roots. The discriminant applies only when a is non-zero.