math

Quadratic Formula Calculator

Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Finds both real and complex roots with step-by-step solutions.

Reviewed by Chase FloiedUpdated April 16, 2026

This free online quadratic formula calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Coefficient a

The coefficient of x² (must not be zero)

Coefficient b

The coefficient of x

Coefficient c

The constant term

Results

Discriminant (Δ)

Root x₁

Root x₂

Vertex x

Vertex y

How to Use This Calculator

Enter your input values

Fill in all required input fields for the Quadratic Formula Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

Read the results

The Quadratic Formula Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

Formula Reference

Quadratic Formula Calculator Formula

See calculator inputs for the governing equation

Variables: All variables and their units are labeled in the calculator interface above. Input fields accept values in multiple unit systems — select your preferred unit from the dropdown next to each field.

When to Use This Calculator

•Use the Quadratic Formula Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
•Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
•Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
•Use it to explore the behavior of mathematical functions across a range of inputs.

About This Calculator

The Quadratic Formula Calculator is a free mathematical calculation tool for students, educators, and professionals who need quick, reliable results. Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. Finds both real and complex roots with step-by-step solutions. The underlying algorithms implement well-established mathematical formulas and numerical methods. Results are computed instantly in the browser. This tool is useful for learning, verification of hand calculations, and rapid exploration of mathematical relationships. All computation happens locally — no data is sent to a server.

About Quadratic Formula Calculator

The Quadratic Formula Calculator solves equations of the form ax² + bx + c = 0 by applying the universal quadratic formula. This ancient method, first formalized by Indian mathematicians around 628 AD and later refined by Persian scholars, remains one of the most important algebraic tools. Every quadratic equation has exactly two solutions (counting multiplicity), which may be real and distinct, real and equal, or complex conjugates. The quadratic formula x = (-b ± √(b² - 4ac)) / 2a provides these solutions directly from the coefficients, making it an indispensable tool in mathematics, physics, engineering, and economics. Whether you are solving projectile motion problems, optimizing business models, or analyzing geometric relationships, understanding the quadratic formula is foundational to algebraic fluency.

The Math Behind It

A quadratic equation is any polynomial equation of degree two, written in standard form as ax² + bx + c = 0, where a ≠ 0. The quadratic formula is derived by completing the square on the general equation. Starting with ax² + bx + c = 0, divide through by a: x² + (b/a)x + c/a = 0. Move the constant: x² + (b/a)x = -c/a. Add (b/2a)² to both sides: (x + b/2a)² = (b² - 4ac)/4a². Take the square root: x + b/2a = ±√(b² - 4ac)/2a. Solve for x to get the formula. The discriminant Δ = b² - 4ac determines the nature of the roots: when Δ > 0, there are two distinct real roots; when Δ = 0, there is one repeated real root; when Δ < 0, the roots are complex conjugates. The sum of the roots equals -b/a and the product equals c/a, known as Vieta's formulas. The vertex of the parabola y = ax² + bx + c occurs at x = -b/(2a), which is the average of the two roots. The parabola opens upward when a > 0 and downward when a < 0. Quadratic equations model parabolic trajectories, optimization problems, area calculations, and many natural phenomena involving squared relationships.

Formula Reference

Quadratic Formula

x = (-b ± √(b² - 4ac)) / 2a

Variables: a = coefficient of x², b = coefficient of x, c = constant term

Discriminant

Δ = b² - 4ac

Variables: Determines the nature and number of roots

Worked Examples

Example 1: Two distinct real roots

Solve 2x² - 7x + 3 = 0

Step 1:Identify a = 2, b = -7, c = 3

Step 2:Calculate discriminant: Δ = (-7)² - 4(2)(3) = 49 - 24 = 25

Step 3:Since Δ > 0, there are two distinct real roots

Step 4:x₁ = (7 + √25) / (2·2) = (7 + 5) / 4 = 12/4 = 3

Step 5:x₂ = (7 - √25) / (2·2) = (7 - 5) / 4 = 2/4 = 0.5

x = 3 or x = 0.5

Example 2: Repeated root

Solve x² - 6x + 9 = 0

Step 1:Identify a = 1, b = -6, c = 9

Step 2:Calculate discriminant: Δ = (-6)² - 4(1)(9) = 36 - 36 = 0

Step 3:Since Δ = 0, there is one repeated root

Step 4:x = -(-6) / (2·1) = 6/2 = 3

x = 3 (double root)

Example 3: Complex roots

Solve x² + 2x + 5 = 0

Step 1:Identify a = 1, b = 2, c = 5

Step 2:Calculate discriminant: Δ = 4 - 20 = -16

Step 3:Since Δ < 0, roots are complex

Step 4:x = (-2 ± √(-16)) / 2 = (-2 ± 4i) / 2

Step 5:x₁ = -1 + 2i, x₂ = -1 - 2i

x = -1 ± 2i

Common Mistakes & Tips

!Forgetting that 'a' must not be zero - if a = 0 the equation is linear, not quadratic
!Sign errors when substituting negative values of b into -b ± √(b² - 4ac)
!Forgetting to divide the entire numerator by 2a, not just part of it
!Confusing the discriminant sign: negative discriminant means complex roots, not 'no solution'
!Not simplifying the final answer when the square root yields a perfect square

Related Concepts

Discriminant

Analyze the nature of quadratic roots without solving

Completing the Square

An alternative method for solving quadratics and deriving vertex form

Factoring Trinomials

Factor quadratic expressions when integer roots exist

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Completing the Square Calculator

Complex Number Calculator

Cubic Equation Calculator

Diamond Problem Calculator

Discriminant Calculator

Factoring Trinomials Calculator

Frequently Asked Questions

Can the quadratic formula solve any quadratic equation?

Yes. The quadratic formula works for all quadratic equations ax² + bx + c = 0 where a ≠ 0, whether the roots are real or complex. It is a universal method.

What does it mean when the discriminant is negative?

A negative discriminant means the equation has no real solutions. Instead, it has two complex conjugate roots of the form p ± qi, where i = √(-1).

How is the quadratic formula related to the vertex?

The x-coordinate of the vertex is -b/(2a), which is the midpoint between the two roots. The ± part of the formula shows how far each root is from the vertex.