Quadratic Formula Calculator Roots

The quadratic formula is a fundamental tool in algebra for finding the roots of quadratic equations. This calculator quickly computes the roots of any quadratic equation in the standard form ax² + bx + c = 0, providing both real and complex solutions when needed.

What is the Quadratic Formula?

The quadratic formula is a formula that provides the solutions to the equation ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The formula is derived from completing the square and is one of the most important formulas in algebra.

Quadratic Formula

The quadratic formula is written as:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

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

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

If the discriminant is positive, there are two distinct real roots
If the discriminant is zero, there is exactly one real root (a repeated root)
If the discriminant is negative, there are two complex conjugate roots

The quadratic formula is widely used in physics, engineering, economics, and many other fields where quadratic equations appear.

How to Use This Calculator

Using the quadratic formula calculator is simple:

Enter the coefficients a, b, and c of your quadratic equation in the input fields
Click the "Calculate Roots" button
View the results, which include:

The two roots of the equation
The discriminant value
An interpretation of the results
A graphical representation of the quadratic function

Note: The calculator will display complex roots if the discriminant is negative. Complex roots are expressed in the form a + bi, where i is the imaginary unit.

Quadratic Formula Examples

Let's look at some examples of how to use the quadratic formula:

Example 1: Two Distinct Real Roots

Find the roots of x² - 5x + 6 = 0

Using the quadratic formula:

x = [5 ± √(25 - 24)] / 2 = [5 ± 1] / 2

So the roots are x = 3 and x = 2

Example 2: One Real Root (Repeated Root)

Find the roots of x² - 6x + 9 = 0

Using the quadratic formula:

x = [6 ± √(36 - 36)] / 2 = 6 / 2 = 3

So there is one real root at x = 3

Example 3: Complex Roots

Find the roots of x² + 4x + 5 = 0

Using the quadratic formula:

x = [-4 ± √(16 - 20)] / 2 = [-4 ± √(-4)] / 2 = [-4 ± 2i] / 2

So the roots are x = -1 + i and x = -1 - i

Quadratic Formula FAQ

What is the quadratic formula used for?

The quadratic formula is used to find the roots of quadratic equations, which are equations of the form ax² + bx + c = 0. It's widely used in algebra, physics, engineering, and other fields where quadratic equations appear.

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

A quadratic equation has real roots if the discriminant (b² - 4ac) is positive. If the discriminant is zero, there is exactly one real root. If the discriminant is negative, the equation has complex roots.

What does the discriminant tell you about the roots?

The discriminant (b² - 4ac) provides information about the nature of the roots of a quadratic equation:

Positive discriminant: Two distinct real roots
Zero discriminant: One real root (a repeated root)
Negative discriminant: Two complex conjugate roots

Can the quadratic formula be used for any quadratic equation?

Yes, the quadratic formula can be used for any quadratic equation in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It will always provide the correct roots, whether they are real or complex.