How to Find Discriminant of Quadratic Equation?

The discriminant of a quadratic equation is a single number that tells you — without solving the equation — how many real solutions it has. It comes directly from the quadratic formula and is one of the most efficient tools for analyzing quadratics in Algebra 1 and Algebra 2. This guide explains how to calculate and interpret the discriminant with worked examples and practice problems.

What Is the Discriminant?

The discriminant is the expression under the radical in the quadratic formula:

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\(\color{blue}{D = b^{2} – 4\text{ ac }}\)

For the equation \(\color{blue}{\text{ ax }^{2} + \text{ bx } + c = 0}\), the discriminant tells you the nature of the roots without computing them. Its value can be positive, zero, or negative.

How to Find and Interpret the Discriminant

Case 1: D > 0 — Two Distinct Real Solutions

The quadratic formula produces two different real roots. The parabola crosses the x-axis at two distinct points.

• \(\color{blue}{x^{2} – 5x + 6 = 0}\): \(\color{blue}{D = 25 – 24 = 1 > 0}\) → two real solutions.

Case 2: \(\color{blue}{D = 0}\) — One Real Solution (Double Root)

Both roots are equal. The parabola just touches the x-axis at its vertex and does not cross it.

• \(\color{blue}{x^{2} + 6x + 9 = 0}\): \(\color{blue}{D = 36 – 36 = 0}\) → one repeated root, \(\color{blue}{x = -3}\).

Case 3: D < 0 — No Real Solutions

The square root of a negative number is not real. The parabola does not cross the x-axis at all.

• \(\color{blue}{x^{2} + 2x + 5 = 0}\): \(\color{blue}{D = 4 – 20 = -16 < 0}\) → no real solutions.

Step-by-Step Summary

1. Write the equation in standard form \(\color{blue}{\text{ ax }^{2} + \text{ bx } + c = 0}\).
2. Identify \(\color{blue}{a}\), \(\color{blue}{b}\), and \(\color{blue}{c}\).
3. Calculate \(\color{blue}{D = b^{2} – 4\text{ ac }}\).
4. Interpret: \(\color{blue}{D > 0}\) → 2 real roots; \(\color{blue}{D = 0}\) → 1 real root; \(\color{blue}{D < 0}\) → 0 real roots.

Watch: The Discriminant of a Quadratic Equation

Khan Academy explains how to compute and interpret the discriminant:

Discriminant – Worked Examples

Example 1: Find the discriminant of \(\color{blue}{x^{2} – 5x + 6 = 0}\). How many real solutions?

\(\color{blue}{a=1, b=-5, c=6}\).
\(\color{blue}{D = (-5)^{2} – 4(1)(6) = 25 – 24 = 1}\).
\(\color{blue}{D > 0}\) → two distinct real solutions.

Example 2: Find the discriminant of \(\color{blue}{x^{2} + 6x + 9 = 0}\). How many real solutions?

\(\color{blue}{a=1, b=6, c=9}\).
\(\color{blue}{D = 36 – 36 = 0}\).
\(\color{blue}{D = 0}\) → one repeated real solution: \(\color{blue}{x = -3}\).

Example 3: Find the discriminant of \(\color{blue}{x^{2} + 2x + 5 = 0}\). How many real solutions?

\(\color{blue}{a=1, b=2, c=5}\).
\(\color{blue}{D = 4 – 20 = -16}\).
\(\color{blue}{D < 0}\) → no real solutions.

Example 4: Without solving, determine the number of real solutions to \(\color{blue}{3x^{2} – 7x + 2 = 0}\).

\(\color{blue}{D = 49 – 24 = 25 > 0}\) → two distinct real solutions. (They are \(\color{blue}{x = 2}\) and \(\color{blue}{x = \frac{1}{3}}\).)

More Practice: Discriminant for Types of Solutions

Khan Academy extends the discriminant to classify the nature of roots across multiple examples:

Exercises: Finding the Discriminant

1. Find D and state the number of real solutions: \(\color{blue}{x^{2} – 4x + 4 = 0}\)
2. Find D and state the number of real solutions: \(\color{blue}{2x^{2} + 3x – 5 = 0}\)
3. Find D and state the number of real solutions: \(\color{blue}{x^{2} + x + 1 = 0}\)
4. Find D and state the number of real solutions: \(\color{blue}{4x^{2} – 12x + 9 = 0}\)
5. For what value of \(\color{blue}{k}\) does \(\color{blue}{x^{2} + \text{ kx } + 4 = 0}\) have exactly one real solution?

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Answers

1. \(\color{blue}{D = 16 – 16 = 0}\); one repeated real solution (\(\color{blue}{x = 2}\))
2. \(\color{blue}{D = 9 + 40 = 49 > 0}\); two distinct real solutions
3. \(\color{blue}{D = 1 – 4 = -3 < 0}\); no real solutions
4. \(\color{blue}{D = 144 – 144 = 0}\); one repeated real solution (\(\color{blue}{x = \frac{3}{2}}\))
5. \(\color{blue}{D = k^{2} – 16 = 0}\)  →  \(\color{blue}{k = \pm 4}\)

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Free Finding Discriminant of Quadratic Equation Worksheet

Ready to practice on your own? Download our free Finding Discriminant of Quadratic Equation worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Finding Discriminant of Quadratic Equation before a quiz or test.

Download The Discriminant Worksheet

Frequently Asked Questions

Can the discriminant be used to determine if a quadratic factors over the integers?

Yes. If the discriminant is a perfect square (0, 1, 4, 9, 16, 25, …), the quadratic factors over the integers. If the discriminant is positive but not a perfect square, the roots are irrational.

What does a negative discriminant mean geometrically?

A negative discriminant means the parabola does not intersect the x-axis. The entire parabola is either above or below the x-axis depending on the sign of \(\color{blue}{a}\).

How is the discriminant related to the quadratic formula?

The discriminant \(\color{blue}{D = b^{2} – 4\text{ ac }}\) is the expression inside the radical in the quadratic formula: \(\color{blue}{x = \frac{(-b \pm \sqrt{D})}{(2a)}}\). When \(\color{blue}{D < 0}\), the radical is imaginary, giving no real solutions. When \(\color{blue}{D = 0}\), both solutions collapse to \(\color{blue}{x = -\frac{b}{(2a)}}\).

Related Topics

• Solving a Quadratic Equation
• Factoring Quadratics
• Completing the Square
• Quadratic Function
• Graphing Quadratic Functions