Quadratic Formula Calculator with Roots

This quadratic formula calculator helps you solve quadratic equations and find their roots. Whether you're a student studying algebra or a professional working with mathematical models, this tool provides quick and accurate solutions.

What is the Quadratic Formula?

The quadratic formula is a fundamental tool in algebra used to find the roots of a quadratic equation. A quadratic equation is any equation that can be written in the form:

ax² + bx + c = 0

Where a, b, and c are constants, and a ≠ 0. The roots of the equation are the values of x that satisfy the equation. The quadratic formula provides a direct method to calculate these roots:

x = [-b ± √(b² - 4ac)] / (2a)

The term under the square root (b² - 4ac) is called the discriminant. It 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.

How to Use This Calculator

Using this quadratic formula calculator is simple:

Enter the coefficients a, b, and c from your quadratic equation.
Click the "Calculate" button to find the roots.
View the results, including the roots and discriminant.
Use the optional chart to visualize the quadratic function.

The calculator will display the roots in a clear format and explain the nature of the roots based on the discriminant.

Formula Explanation

The quadratic formula is derived from completing the square method. Here's a step-by-step explanation:

Start with the standard form: ax² + bx + c = 0
Divide all terms by a: x² + (b/a)x + (c/a) = 0
Move the constant term to the other side: x² + (b/a)x = -c/a
Complete the square by adding (b/2a)² to both sides: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
Write the left side as a perfect square: (x + b/2a)² = (b² - 4ac)/4a²
Take the square root of both sides: x + b/2a = ±√(b² - 4ac)/2a
Solve for x: x = [-b ± √(b² - 4ac)] / (2a)

Note: The quadratic formula only works when a ≠ 0. If a = 0, the equation is linear, not quadratic.

Worked Example

Let's solve the quadratic equation 2x² + 4x - 6 = 0 using the quadratic formula.

Step 1: Identify coefficients

a = 2, b = 4, c = -6

Step 2: Calculate discriminant

Discriminant = b² - 4ac = (4)² - 4(2)(-6) = 16 + 48 = 64

Step 3: Apply quadratic formula

x = [-b ± √(b² - 4ac)] / (2a) = [-4 ± √64] / 4 = [-4 ± 8] / 4

Step 4: Find roots

First root: x = (-4 + 8)/4 = 4/4 = 1

Second root: x = (-4 - 8)/4 = -12/4 = -3

The roots of the equation 2x² + 4x - 6 = 0 are x = 1 and x = -3.

Frequently Asked Questions

What is the discriminant in the quadratic formula?

The discriminant is the part of the quadratic formula under the square root (b² - 4ac). It tells you about the nature of the roots:

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

Can the quadratic formula be used for all quadratic equations?

Yes, the quadratic formula can be used for any quadratic equation as long as the coefficient of x² (a) is not zero. If a = 0, the equation becomes linear and should be solved using linear methods.

What if the discriminant is negative?

When the discriminant is negative, the roots are complex numbers. The calculator will display them in the form a ± bi, where i is the imaginary unit (√-1).