Quadratic Equation of Roots Calculator

A quadratic equation is a second-degree polynomial equation in a single variable x with at least one x² term. The general form is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This calculator helps you find the roots of any quadratic equation by applying the quadratic formula.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2. It has the general form:

ax² + bx + c = 0

Where:

a, b, and c are constants
a ≠ 0 (if a = 0, the equation is linear, not quadratic)
x is the variable

The solutions to a quadratic equation are called roots or zeros. A quadratic equation can have:

Two distinct real roots
One real root (a repeated root)
No real roots (the roots are complex numbers)

Quadratic equations are used in many areas of mathematics and science, including physics, engineering, and economics.

The Quadratic Formula

The standard method for solving quadratic equations is the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

a, b, and c are the coefficients from the quadratic equation
√(b² - 4ac) is called the discriminant
The ± symbol indicates there are two solutions

The discriminant (b² - 4ac) determines the nature of the roots:

If b² - 4ac > 0: Two distinct real roots
If b² - 4ac = 0: One real root (repeated)
If b² - 4ac < 0: No real roots (complex roots)

Note: The quadratic formula will always give you the correct roots, regardless of the values of a, b, and c.

How to Use This Calculator

Using our quadratic equation calculator is simple:

Enter the coefficients a, b, and c in the input fields
Click the "Calculate" button
View the results, including the roots and discriminant
Use the chart to visualize the quadratic function

The calculator will:

Calculate both roots using the quadratic formula
Determine the discriminant and its meaning
Display the quadratic function in standard form
Show a graph of the quadratic function

Example Calculation

For the equation x² - 5x + 6 = 0:

a = 1, b = -5, c = 6
Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1
Roots: x = [5 ± √1]/2 = 3 and 2

Examples of Quadratic Equations

Here are some examples of quadratic equations and their solutions:

Example 1: Two Distinct Real Roots

Equation: 2x² - 4x - 6 = 0

Solution:

a = 2, b = -4, c = -6
Discriminant = (-4)² - 4(2)(-6) = 16 + 48 = 64
Roots: x = [4 ± √64]/4 = [4 ± 8]/4
x₁ = (4 + 8)/4 = 3
x₂ = (4 - 8)/4 = -1

Example 2: One Real Root (Repeated)

Equation: x² - 6x + 9 = 0

Solution:

a = 1, b = -6, c = 9
Discriminant = (-6)² - 4(1)(9) = 36 - 36 = 0
Root: x = [6 ± √0]/2 = 3 (repeated)

Example 3: No Real Roots (Complex)

Equation: x² + 2x + 5 = 0

Solution:

a = 1, b = 2, c = 5
Discriminant = 2² - 4(1)(5) = 4 - 20 = -16
Roots: x = [-2 ± √-16]/2 = [-2 ± 4i]/2
x₁ = -1 + 2i
x₂ = -1 - 2i

Frequently Asked Questions

What is the difference between a linear and quadratic equation?

A linear equation has a highest power of 1 (e.g., y = mx + b), while a quadratic equation has a highest power of 2 (e.g., ax² + bx + c = 0). Quadratic equations can have two solutions, while linear equations have only one.

How do I know if a quadratic equation has real roots?

A quadratic equation has real roots if the discriminant (b² - 4ac) is greater than or equal to zero. If the discriminant is negative, the roots are complex numbers.

Can the quadratic formula be used for any quadratic equation?

Yes, the quadratic formula can be used to solve any quadratic equation, regardless of the values of a, b, and c. It's the most reliable method for finding the roots of a quadratic equation.

What is the vertex of a quadratic equation?

The vertex of a quadratic equation is the point where the parabola represented by the equation reaches its maximum or minimum value. The x-coordinate of the vertex is given by -b/(2a).

How can I graph a quadratic equation?

To graph a quadratic equation, you can use the vertex form (y = a(x-h)² + k) or plot points using the quadratic formula. Our calculator includes a graph visualization to help you understand the shape of the quadratic function.