Quadratic Equation Solver

Enter coefficients a, b, and c to solve ax² + bx + c = 0. Instantly find real or complex roots, discriminant, vertex, and axis of symmetry.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter Coefficient 'a'
Input the coefficient of the x² term in your quadratic equation (ax² + bx + c = 0). This value cannot be zero.
2
Enter Coefficient 'b'
Provide the coefficient of the x term in your quadratic equation.
3
Enter Constant Term 'c'
Input the constant term in your quadratic equation.
4
Review Roots and Parabola Properties
The calculator will instantly display both roots of the equation, the discriminant, the vertex coordinates, the axis of symmetry, and the direction the parabola opens.

Example Calculation

A mathematician needs to find the roots of the quadratic equation x² − 5x + 6 = 0.

a (x² coefficient)

b (x coefficient)

-5

c (constant term)

Results

Tips

Check the Discriminant First

Before solving for roots, calculate the discriminant (b² - 4ac). A positive discriminant means two real roots, zero means one repeated real root, and a negative discriminant means two complex (imaginary) roots.

Identify Parabola Direction

The sign of the 'a' coefficient dictates the parabola's opening: a positive 'a' means the parabola opens upward, indicating a minimum point, while a negative 'a' means it opens downward, indicating a maximum point.

Vertex is Key to Extremum

The vertex of the parabola, given by (-b/2a, f(-b/2a)), represents the maximum or minimum point of the quadratic function. Understanding this helps in optimization problems, like finding the maximum height of a projectile or minimum cost.

Solving Quadratic Equations for Real and Complex Roots

The Quadratic Equation Solver provides an immediate solution to any equation in the standard form ax² + bx + c = 0. This powerful tool delivers not only the roots (x-intercepts) but also critical insights into the parabola's characteristics, such as the discriminant, vertex, and axis of symmetry. For example, solving x² - 5x + 6 = 0 reveals two real roots, x₁ = 3 and x₂ = 2, indicating where the parabola crosses the x-axis. This fundamental mathematical operation is indispensable across physics, engineering, and economics for modeling various phenomena.

The Significance of Quadratic Roots

Understanding the roots of a quadratic equation is crucial because they represent the values of the variable that satisfy the equation, or graphically, the x-intercepts of the corresponding parabola. In real-world applications, these roots often signify critical points. For instance, in physics, the roots of a quadratic equation modeling projectile motion might indicate when an object hits the ground (height = 0). In economics, they could represent break-even points where profit is zero. The nature of these roots—whether real, repeated, or complex—provides immediate insight into the problem's feasibility and the behavior of the modeled system.

Unpacking the Quadratic Formula Behind the Solver

The Quadratic Equation Solver uses the well-known quadratic formula, which directly provides the roots (x-values) for any equation in the form ax² + bx + c = 0.

The formula is:

x = (-b ± √(b² - 4ac)) / (2a)

Where:

a is the coefficient of x²
b is the coefficient of x
c is the constant term
The term b² - 4ac is known as the discriminant.

Additionally, the vertex (h, k) and axis of symmetry (x = h) are calculated as:

vertex x (h) = -b / (2a)
vertex y (k) = a × h² + b × h + c
axis of symmetry = x = h

💡 To delve into other geometric calculations involving rotations, our Volume by Disk Method Calculator can help visualize and quantify complex shapes.

Solving x² − 5x + 6 = 0 Step by Step

Let's use the Quadratic Equation Solver to find the properties of the equation x² − 5x + 6 = 0.

Identify Coefficients:
- a = 1
- b = -5
- c = 6
Calculate the Discriminant:
- Discriminant = b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1
- Since the discriminant (1) is positive, there will be two distinct real roots.
Calculate the Roots using the Quadratic Formula:
- x₁ = (-(-5) + √1) / (2 * 1) = (5 + 1) / 2 = 6 / 2 = 3
- x₂ = (-(-5) - √1) / (2 * 1) = (5 - 1) / 2 = 4 / 2 = 2
- So, Root 1 is 3, and Root 2 is 2.
Calculate the Vertex:
- Vertex x = -b / (2a) = -(-5) / (2 * 1) = 5 / 2 = 2.5
- Vertex y = (1)(2.5)² + (-5)(2.5) + 6 = 6.25 - 12.5 + 6 = -0.25
- The Vertex is (2.5, -0.25).
Determine Axis of Symmetry:
- The axis of symmetry is x = 2.5.
Parabola Direction:
- Since 'a' (1) is positive, the parabola opens Upward.

💡 For exploring other mathematical concepts involving volumes of revolution, our Volume by Shell Method Calculator offers further insights into calculus applications.

The Significance of Quadratic Roots

The Enduring Legacy of the Quadratic Formula

The quadratic formula, a cornerstone of algebra, boasts a rich history spanning millennia. Early forms of solving quadratic equations appeared in ancient Babylonian texts around 2000 BCE, primarily through geometric methods. Indian mathematicians like Brahmagupta in the 7th century CE provided explicit solutions, including negative roots. However, the modern algebraic formulation, expressed as x = (-b ± √(b² - 4ac)) / (2a), is largely attributed to the Persian mathematician Muhammad ibn Musa al-Khwarizmi in the 9th century and later refined by European mathematicians such as René Descartes in the 17th century. Its standardization allowed for a universal method to solve any quadratic equation, becoming an indispensable tool in mathematics, science, and engineering for modeling everything from financial growth to the trajectory of objects.

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. Its standard form is ax² + bx + c = 0, where 'a', 'b', and 'c' are coefficients, and 'x' is the variable. The coefficient 'a' cannot be zero, as that would make it a linear equation. These equations are fundamental in algebra and have wide applications in science and engineering.

What does the discriminant (b²−4ac) tell you about the roots?

The discriminant, calculated as b² - 4ac, provides critical information about the nature of a quadratic equation's roots without actually solving for them. If the discriminant is positive (>0), there are two distinct real roots. If it is zero (=0), there is exactly one real root (a repeated root). If the discriminant is negative (<0), there are two complex (imaginary) roots, which are conjugates of each other. This value determines whether the parabola intersects the x-axis, touches it, or does not intersect it at all.

How do the roots of a quadratic equation relate to its graph?

The real roots of a quadratic equation (ax² + bx + c = 0) correspond to the x-intercepts of its parabolic graph. If there are two distinct real roots, the parabola crosses the x-axis at two different points. If there is one repeated real root, the parabola touches the x-axis at exactly one point (its vertex). If there are two complex roots, the parabola does not intersect the x-axis at all. The roots indicate the values of x for which the quadratic function equals zero.