Our free Quadratic Equation Calculator helps you solve quadratic equations quickly and accurately. Enter the coefficients a, b, and c to find the roots of an equation in the standard form ax² + bx + c = 0. Whether you are a student, teacher, or mathematics enthusiast, this calculator makes solving quadratic equations simple and convenient.

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two. It contains a variable raised to the power of two and is generally written in the following standard form:

ax² + bx + c = 0

Here, a, b, and c are constants, and a ≠ 0. The variable x represents the unknown value. The solutions of a quadratic equation are called its roots, which are the values of x that make the equation equal to zero.

Quadratic Formula

The quadratic formula is a standard method for finding the roots of a quadratic equation. It works for equations that can be expressed in the form ax² + bx + c = 0.

x = (-b ± √(b² - 4ac)) / 2a

The expression b² - 4ac is known as the discriminant. It helps determine the type and number of roots of the equation.

Types of Roots Based on the Discriminant

• Discriminant greater than zero: The equation has two distinct real roots.
• Discriminant equal to zero: The equation has two equal real roots, or one repeated root.
• Discriminant less than zero: The equation has two complex conjugate roots and no real roots.

How to Use the Quadratic Equation Calculator

Follow these simple steps to calculate the roots of a quadratic equation:

1. Enter the value of coefficient a.
2. Enter the value of coefficient b.
3. Enter the constant value c.
4. Click the calculate button to solve the equation.
5. Review the calculated roots and other results displayed by the calculator.

Make sure that the value of a is not zero, because an equation with a = 0 is not a quadratic equation.

Example of Solving a Quadratic Equation

Consider the following quadratic equation:

x² - 5x + 6 = 0

In this equation:

• a = 1
• b = -5
• c = 6

First, calculate the discriminant:

D = b² - 4ac

D = (-5)² - 4 × 1 × 6 = 25 - 24 = 1

Next, apply the quadratic formula:

x = (5 ± √1) / 2

x = (5 ± 1) / 2

Therefore, the two roots are:

• x₁ = 3
• x₂ = 2

The solutions of the equation x² - 5x + 6 = 0 are 2 and 3.

Benefits of Using an Online Quadratic Equation Calculator

• Quick results: Calculate quadratic roots in seconds.
• Improved accuracy: Reduce common arithmetic mistakes when applying the quadratic formula.
• Easy to use: Simply enter the three coefficients to solve an equation.
• Understand the formula: Use worked examples to learn how quadratic equations are solved.
• Useful for learning: Practice algebra problems for school, college, and mathematics exams.
• Accessible online: Solve equations without installing additional software.

Frequently Asked Questions (FAQs)

1. What is a quadratic equation calculator?

A quadratic equation calculator is an online tool that solves equations in the form ax² + bx + c = 0. It uses the quadratic formula to calculate the roots based on the values of a, b, and c.

2. What is the formula for solving a quadratic equation?

The quadratic formula is x = (-b ± √(b² - 4ac)) / 2a. Substituting the coefficients into this formula gives the solutions of the equation when a is not zero.

3. What is the discriminant of a quadratic equation?

The discriminant is the expression b² - 4ac. It indicates whether a quadratic equation has two distinct real roots, one repeated real root, or two complex conjugate roots.

4. Can a quadratic equation have two solutions?

Yes. A quadratic equation has two roots when solutions are counted with multiplicity. Depending on the discriminant, the roots may be two distinct real numbers, equal real numbers, or complex conjugates.

5. Can a quadratic equation have no real roots?

Yes. If the discriminant is negative, the equation has no real roots. Instead, it has two complex conjugate roots.

6. What happens if the coefficient a is zero?

If a = 0, the equation is no longer quadratic because the x² term disappears. It becomes a linear equation or a constant equation, depending on the values of b and c.

7. Is this Quadratic Equation Calculator free to use?

Yes, you can use this free online Quadratic Equation Calculator to solve quadratic equations quickly without performing every calculation manually.