Quick Answer
The discriminant is b² − 4ac from the quadratic formula. If it is positive, the equation has two distinct real solutions. If it is zero, the equation has exactly one repeated real solution, also called a double root. If it is negative, there are no real solutions, although there are two complex solutions.
What Is the Discriminant?
The discriminant is the expression b² − 4ac, which appears under the square root sign in the quadratic formula:
x = (−b ± √(b² − 4ac)) / (2a)
For a quadratic equation ax² + bx + c = 0, the discriminant tells you how many real solutions exist — before you do any of the work of solving.
The symbol for the discriminant is Δ (the Greek letter delta), so you may see it written as Δ = b² − 4ac.
| Discriminant | Solutions | X-intercepts |
|---|---|---|
| b² − 4ac > 0 | Two distinct real solutions | Two x-intercepts |
| b² − 4ac = 0 | One repeated real solution | One x-intercept |
| b² − 4ac < 0 | No real solutions | No x-intercepts |
The Three Cases
Case 1 — Discriminant > 0 (positive):
The square root of a positive number is real, and the ± gives two different values. The equation has TWO distinct real solutions.
Case 2 — Discriminant = 0:
The square root of zero is zero. The ± makes no difference: both solutions are the same. The equation has EXACTLY ONE real solution (called a repeated or double root).
Case 3 — Discriminant < 0 (negative):
You cannot take the square root of a negative number in the real number system. The equation has NO real solutions. (In higher math, these become two complex solutions involving i = √(−1).)
How to Calculate the Discriminant
Step 1: Write the equation in standard form ax² + bx + c = 0.
Step 2: Identify a, b, and c.
Step 3: Compute b² − 4ac.
Step 4: Interpret the sign.
That is all. You do not need to finish the quadratic formula — just compute the discriminant and read the result.
Graphical Meaning
The discriminant also describes how the parabola y = ax² + bx + c intersects the x-axis:
Discriminant > 0: the parabola crosses the x-axis at TWO points.
Discriminant = 0: the parabola touches the x-axis at exactly ONE point (the vertex sits on the x-axis).
Discriminant < 0: the parabola does NOT cross the x-axis at all (it floats entirely above or below).
This is a useful visual check: if you can see the graph, count the x-intercepts to predict the discriminant's sign.
Worked Examples
Example 1
How many real solutions does x² − 5x + 6 = 0 have?
Solution
- Identify: a = 1, b = −5, c = 6.
- Discriminant = b² − 4ac = (−5)² − 4(1)(6) = 25 − 24 = 1.
- Discriminant = 1 > 0 → two distinct real solutions.
- (Solving: x = (5 ± 1)/2 → x = 3 or x = 2.)
Answer: Two real solutions: x = 3 and x = 2.
Example 2
How many real solutions does x² − 6x + 9 = 0 have?
Solution
- Identify: a = 1, b = −6, c = 9.
- Discriminant = (−6)² − 4(1)(9) = 36 − 36 = 0.
- Discriminant = 0 → exactly one real solution (double root).
- (Solving: x = 6/2 = 3.)
Answer: One real solution: x = 3 (double root).
Example 3
How many real solutions does 2x² + x + 3 = 0 have?
Solution
- Identify: a = 2, b = 1, c = 3.
- Discriminant = (1)² − 4(2)(3) = 1 − 24 = −23.
- Discriminant = −23 < 0 → no real solutions.
Answer: No real solutions.
Common Mistakes
Forgetting to write the equation in standard form first. If the equation is 3x² = 2x − 1, rewrite it as 3x² − 2x + 1 = 0 before identifying a, b, c.
Sign errors with b². If b = −5, then b² = (−5)² = 25, not −25. Squaring always gives a non-negative result.
Confusing the discriminant with the full quadratic formula. The discriminant is only b² − 4ac — you do not divide by 2a.
Concluding "no solution" when the discriminant is negative. There are no REAL solutions, but complex solutions exist. In Algebra 1, "no real solutions" is the correct phrasing.
Practice Problems
Find the discriminant of x² + 4x + 4 = 0. How many real solutions?
Hint: a = 1, b = 4, c = 4.
Find the discriminant of 3x² − 2x − 1 = 0. How many real solutions?
Hint: a = 3, b = −2, c = −1.
Find the discriminant of x² + 2x + 5 = 0. How many real solutions?
Hint: a = 1, b = 2, c = 5.
For what value of k does x² + kx + 9 = 0 have exactly one real solution?
Hint: Set b² − 4ac = 0 and solve for k.
Practice Problem Answers
Frequently Asked Questions
What can the discriminant tell you?
The discriminant tells you the number and type of solutions of a quadratic equation. It also tells you how many times the graph crosses or touches the x-axis.
What does b² − 4ac tell you?
The expression b² − 4ac is the discriminant. If it is positive, there are two distinct real solutions. If it is zero, there is one repeated real solution. If it is negative, there are no real solutions.
Does the discriminant tell you the actual solutions?
No. The discriminant tells you the number and type of solutions, but you must use factoring, completing the square, or the quadratic formula to find their actual values.
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