🔍 Algebra Solver

Discriminant Calculator - Find the Roots

Q: What is the discriminant of a quadratic equation?

The discriminant is the value $\Delta = b^2 - 4ac$ for a quadratic equation $ax^2 + bx + c = 0$. It determines the number and nature of the roots.

Q: How do you find the discriminant?

Write the quadratic equation in standard form, identify a, b, and c, then substitute them into $\Delta = b^2 - 4ac$ and calculate the result.

Q: What does a positive discriminant mean?

A positive discriminant means the quadratic equation has two distinct real roots.

Q: What does a discriminant of zero mean?

When $\Delta = 0$, the quadratic equation has one repeated real root.

Q: What does a negative discriminant mean?

A negative discriminant means the quadratic equation has two complex conjugate roots and no real roots.

Use this discriminant calculator to find Δ = b² - 4ac and determine the number and nature of roots of a quadratic equation.

Quadratic Coefficients

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Coefficient (a) ℹ️ Info

Coefficient (b) ℹ️ Info

Constant (c) ℹ️ Info

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How to Use & Educational Guide

In-depth guide for Discriminant Calculator

Educational Overview

What Is the Discriminant (Δ = b² - 4ac)?

The discriminant, represented by $\Delta$, is the value calculated from the coefficients of a quadratic equation using $\Delta = b^2 - 4ac$. It tells you the number and nature of the roots without requiring you to solve the entire equation. A discriminant calculator uses this formula to quickly find the value of $\Delta$ and classify the roots.

Discriminant Formula for a Quadratic Equation

For a quadratic equation written in standard form, identify $a$, $b$, and $c$, then substitute them into the discriminant formula:

$$\Delta = b^2 - 4ac$$

The value of the discriminant determines whether the quadratic has two distinct real roots, one repeated real root, or two complex roots.

How to Find the Discriminant Step-by-Step

Step 1 (Write the Quadratic in Standard Form): Rearrange the equation into $ax^2 + bx + c = 0$.
Step 2 (Identify the Coefficients): Find the values of $a$, $b$, and $c$, including their signs.
Step 3 (Use the Discriminant Formula): Substitute the coefficients into $\Delta = b^2 - 4ac$ and calculate the result.
Step 4 (Determine the Roots): Use whether $\Delta$ is positive, zero, or negative to determine the number and nature of the roots.

Discriminant Cases & Worked Examples

Case 1 (Δ > 0): Two Distinct Real Roots — $x^2 - 5x + 6 = 0$

• Step 1 (Identify coefficients): $a = 1$, $b = -5$, and $c = 6$.

• Step 2 (Calculate the discriminant): $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = \mathbf{1}$.

• Step 3 (Interpret the value): Since $\Delta > 0$, the equation has two distinct real roots.

👉 Discriminant: $\Delta = 1$ — 2 Distinct Real Roots

Case 2 (Δ = 0): One Repeated Real Root — $x^2 + 4x + 4 = 0$

• Step 1 (Identify coefficients): $a = 1$, $b = 4$, and $c = 4$.

• Step 2 (Calculate the discriminant): $\Delta = (4)^2 - 4(1)(4) = 16 - 16 = \mathbf{0}$.

• Step 3 (Interpret the value): Since $\Delta = 0$, the equation has one repeated real root.

👉 Discriminant: $\Delta = 0$ — 1 Repeated Real Root

Case 3 (Δ < 0): Two Complex Roots — $x^2 + 2x + 5 = 0$

• Step 1 (Identify coefficients): $a = 1$, $b = 2$, and $c = 5$.

• Step 2 (Calculate the discriminant): $\Delta = (2)^2 - 4(1)(5) = 4 - 20 = \mathbf{-16}$.

• Step 3 (Interpret the value): Since $\Delta < 0$, the equation has two complex conjugate roots.

👉 Discriminant: $\Delta = -16$ — 2 Complex Roots

What Does the Discriminant Tell You About the Roots?

The sign of the discriminant provides a quick way to determine the nature of the roots of a quadratic equation. A positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives two complex conjugate roots.

Δ > 0

Two distinct real roots.

Δ = 0

One repeated real root.

Δ < 0

Two complex conjugate roots.

Applications of the Discriminant

📐 Quadratic Equations & Parabolas

The discriminant shows whether a parabola intersects the x-axis at two points, touches it once, or does not intersect it at all.

⚙️ Engineering & Applied Mathematics

Discriminants help determine whether mathematical models produce distinct real solutions, repeated solutions, or complex solutions.

Common Mistakes to Avoid

Squaring Negative b Incorrectly: If $b = -5$, then $b^2 = (-5)^2 = 25$, not $-25$.
Sign Errors in -4ac: If $c$ is negative, carefully substitute its sign into $-4ac$. For example, $-4(1)(-6) = +24$.
Using the Wrong Coefficients: The equation must be written as $ax^2 + bx + c = 0$ before identifying $a$, $b$, and $c$.
Confusing the Discriminant With the Roots: The discriminant is a value used to determine root nature; it is not itself a root of the equation.

Mathematical accuracy verified by AptCalc Engine