The discriminant of a quadratic equation ax2 + bx + c = 0 is the number b2 − 4ac. Its sign tells how many real solutions the equation has without solving it: positive means two, zero means one (a repeated root), and negative means none, because the parabola never touches the x-axis.
Compute one number, b2 − 4ac, and you know how many times the parabola meets the x-axis: 0, 1 or 2.
Formal definition and notation
For ax2 + bx + c = 0 with real coefficients and a ≠ 0, the discriminant is D = b2 − 4ac, the radicand in the quadratic formula. If D > 0 there are two distinct real roots; if D = 0 there is one real root of multiplicity 2, x = −b2a; if D < 0 there are two non-real complex conjugate roots. When a, b, c are integers, the roots are rational exactly when D is a perfect square.
Symbol
Read it as
Meaning
D or Δ
D, or delta
common names for the discriminant
b2 − 4ac
b squared minus four a c
the formula, using the standard-form coefficients
Key facts
b2 is never negative, so D can only be negative when 4ac is positive, meaning a and c have the same sign.
If a and c have opposite signs, D is positive and there are always two real solutions.
A perfect-square D (0, 1, 4, 9, 16, …) with integer coefficients means the quadratic factors over the integers.
The discriminant only counts and classifies solutions; the quadratic formula finds them.
With D < 0, the solutions involve √D = i√|D|, a complex number.
Discriminant formula
Discriminant formula
D = b2 − 4ac
D > 0
two different real solutions; the parabola crosses the x-axis twice
D = 0
one real solution; the vertex touches the x-axis
D < 0
no real solutions (two complex ones); the parabola misses the x-axis
Write the equation in standard form ax2 + bx + c = 0 before reading a, b and c.
Discriminant examples, step by step
Example 1: Count the solutions
Warm-up
How many real solutions does 2x2 + 3x − 4 = 0 have?
a = 2, b = 3, c = −4.
D = 32 − 4(2)(−4) = 9 + 32 = 41.
41 > 0, so there are two real solutions.
Answer: D = 41, so there are two real solutions.
Example 2: A repeated root
Core
Find the discriminant of 4x2 − 20x + 25 = 0 and describe the solutions.
The quadratic formula gives the solutions themselves. The discriminant is just the piece under its square root, which tells how many there are.
Discriminantvs.Determinant
A determinant is a number computed from a square matrix, such as ad − bc for a 2 by 2 matrix. Different word, different object.
Quick check: 3 questions
Question 1 of 3
What is the discriminant of 3x2 − 2x + 1 = 0?
Show answer and explanation
Answer: B.(−2)2 − 4(3)(1) = 4 − 12 = −8. 16 added 12 to 4, −16 treated (−2)2 as −4, and 8 dropped the sign of the result.
Question 2 of 3
A quadratic has discriminant 49. Which describes its solutions?
Show answer and explanation
Answer: D. 49 is positive, so there are two real solutions, and 49 is a perfect square, so √49 = 7 makes them rational (with integer coefficients). Zero would give one solution, and a positive non-square would give irrational ones.
Question 3 of 3
For which value of c does x2 + 4x + c = 0 have no real solutions?
Show answer and explanation
Answer: A.D = 16 − 4c is negative when c > 4, so c = 5 gives D = −4. c = 4 gives D = 0 (one solution), and 0 and −5 give positive D.
Discriminant: frequently asked questions
What does the discriminant tell you?
It tells how many real solutions a quadratic equation has and what kind they are. Positive: two different real solutions, rational if the discriminant is a perfect square. Zero: one repeated real solution. Negative: no real solutions, only two complex ones. On the graph, it tells whether the parabola crosses, touches or misses the x-axis.
What is the discriminant formula?
For ax2 + bx + c = 0, the discriminant is b2 − 4ac. It is the expression under the square root sign in the quadratic formula x = −b ± √b2 − 4ac2a. Write the equation in standard form first, then substitute the coefficients with their signs.
Can the discriminant be negative?
Yes. A negative discriminant means the quadratic equation has no real solutions, because a real number cannot be the square root of a negative number. For x2 + 1 = 0, D = −4. The equation still has two complex solutions, x = i and x = −i, which you study in Algebra 2.
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