Online Quadratic Roots Calculator

A quadratic equation is a second-degree polynomial equation in a single variable x with three coefficients. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The roots of a quadratic equation are the values of x that satisfy the equation.

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 becomes linear)
x is the variable

Quadratic equations can be solved using various methods, including factoring, completing the square, and using the quadratic formula. The roots of the equation are the solutions that satisfy the equation.

How to Find the Roots of a Quadratic Equation

There are several methods to find the roots of a quadratic equation:

Factoring: Express the quadratic as a product of two binomials.
Completing the Square: Rewrite the equation in the form (x + p)² = q.
Quadratic Formula: Use the formula x = [-b ± √(b² - 4ac)] / (2a).

The quadratic formula is the most general method and works for any quadratic equation, regardless of whether it can be factored.

The Quadratic Formula

The quadratic formula provides the roots of any quadratic equation in the form ax² + bx + c = 0:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

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

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 (a repeated root)
If b² - 4ac < 0: Two complex conjugate roots

Example Calculation

Let's solve the quadratic equation x² - 5x + 6 = 0 using the quadratic formula.

x = [5 ± √(25 - 24)] / 2

x = [5 ± √1] / 2

x = [5 ± 1] / 2

This gives two solutions:

x = (5 + 1)/2 = 3
x = (5 - 1)/2 = 2

Therefore, the roots of the equation x² - 5x + 6 = 0 are x = 2 and x = 3.

Frequently Asked Questions

What is the difference between a linear and quadratic equation?

A linear equation has a degree of 1 (e.g., y = mx + b), while a quadratic equation has a degree of 2 (e.g., ax² + bx + c = 0). Quadratic equations can have two solutions, while linear equations have 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 find the roots of any quadratic equation, regardless of whether it can be factored. It's the most general method for solving quadratic equations.

What does it mean if a quadratic equation has a discriminant of zero?

A discriminant of zero means the quadratic equation has exactly one real root (a repeated root). The parabola touches the x-axis at one point.