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Calculator Answers

Quadratic Formula Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

A quadratic formula calculator is an algebraic tool designed to solve second-degree polynomial equations using the quadratic formula. By receiving the coefficients of the quadratic term, linear term, and constant term, the calculator determines the discriminant to identify the nature of the roots. It then evaluates both solutions, including real numbers or complex conjugate pairs when the discriminant is negative. Students, teachers, and engineers use it to verify solutions, analyze parabolic functions, and solve physical motion equations.

Enter the coefficients a, b, and c for a quadratic equation in the form ax² + bx + c = 0. We compute the discriminant, identify the root type, and return both roots with a step-by-step breakdown that shows the formula applied to your numbers.

See also: Distance Formula Calculator, Midpoint Formula Calculator, and Slope Calculator.

Quick Answer

Solve second-degree quadratic equations of the form ax² + bx + c = 0. Enter coefficients a, b, and c to find real or complex roots instantly.

Coefficients

Enter a, b, and c for the equation ax² + bx + c = 0.

Must be nonzero. · e.g. 1

e.g. -5

e.g. 6

Step by step

  1. 1. Identify a, b, and c. a = 1, b = -5, c = 6.
  2. 2. Calculate the discriminant. D = -5² − 4·1·6 = 25 − 24 = 1.
  3. 3. Plug into the formula. x = (−(-5) ± √1) / 2·1
  4. 4. Simplify. x₁ = 3 and x₂ = 2.
Two real roots

Roots

3, 2

Two distinct real roots from D = 1

Equationx² − 5x + 6 = 0
Discriminant (b² − 4ac)1
x₁3
x₂2

The discriminant D = b² − 4ac determines the root type. Positive D gives two real roots, zero gives a repeated root, negative gives a complex conjugate pair.

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Examples

x² − 5x + 6 = 0

Two real roots · x = 2, 3

x² + 2x + 1 = 0

Repeated root · x = −1

x² + 2x + 5 = 0

Complex roots · x = −1 ± 2i

2x² − 4x − 6 = 0

Two real roots · x = −1, 3

How it works

For any equation in the standard form ax² + bx + c = 0 with a ≠ 0, the two roots come from the quadratic formula:

Roots · x = (−b ± √(b² − 4ac)) / 2a

Discriminant · D = b² − 4ac

The discriminant tells you what kind of roots to expect:

  • D > 0 two distinct real roots
  • D = 0 one repeated real root (double root)
  • D < 0 two complex roots in a conjugate pair

Learn the concept

These guides explain the idea this tool computes, with worked examples and the distinctions readers most often trip over.

What Is a Variable in Math? Solving for the variable, and the a, b, c naming convention.
Read the guide: Quadratic Formula explains what each part of the formula means, how the discriminant works, and walks through three examples (two real, one repeated, two complex).

Methods for solving quadratic equations

Depending on the structure of the equation, different algebraic methods can be used to find its roots:

  • Factoring: Rewriting the trinomial as a product of binomials (e.g., (x − 2)(x − 3) = 0). This is the fastest approach when the roots are small, simple integers.
  • Quadratic Formula: Applying the universal equation x = (−b ± √(b² − 4ac)) / 2a. This is the most reliable method because it works for all coefficients, including decimal, irrational, or complex numbers.
  • Completing the Square: Rearranging terms to create a perfect square trinomial on one side of the equation. This method is mathematically foundational and is actually used to derive the quadratic formula itself.

The discriminant (b² − 4ac)

The discriminant is the expression b² − 4ac located under the square root sign in the quadratic formula. It determines the number and nature of the equation's solutions. The calculator above calculates and reports the discriminant value on every solve.

Discriminant ConditionRoot TypeExample Equation & Value
D > 0Two distinct real rootsx² − 5x + 6 = 0 (D = 1)
D = 0One repeated real root (double root)x² − 4x + 4 = 0 (D = 0)
D < 0Two complex roots (conjugate pair)x² + x + 1 = 0 (D = −3)

Worked example

Let's solve the quadratic equation x² − 5x + 6 = 0 step-by-step using the quadratic formula:

  1. 1. Identify coefficients: Here, a = 1, b = −5, and c = 6.
  2. 2. Calculate the discriminant:
    D = b² − 4ac = (−5)² − 4(1)(6) = 25 − 24 = 1.
    Since D > 0, the equation has two distinct real solutions.
  3. 3. Plug into the quadratic formula:
    x = (−b ± √D) / 2a = (−(−5) ± √1) / 2(1) = (5 ± 1) / 2.
  4. 4. Simplify the solutions:
    * x₁ = (5 + 1) / 2 = 6 / 2 = 3
    * x₂ = (5 − 1) / 2 = 4 / 2 = 2
    The two solutions are x = 3 and x = 2.

Related math calculators

Sources: The quadratic formula and standard quadratic equation solving methods are part of the core high school algebra curriculum (Common Core State Standards HSA-REI.B.4). Last reviewed: June 2026.

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Frequently asked questions

The quadratic formula solves any equation in the form ax² + bx + c = 0. The two roots are x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant; it tells you whether the roots are real and distinct, real and repeated, or complex.

Identify the three coefficients a, b, and c from your equation. Compute the discriminant b² − 4ac. If it is positive, take its square root and use the formula to find two real roots. If it is zero, the formula gives one repeated root. If it is negative, the roots are a complex conjugate pair.

The discriminant (D = b² − 4ac) describes how many real roots the equation has. D > 0 means two distinct real roots, D = 0 means one repeated real root, and D < 0 means no real roots (the two roots are complex conjugates).

The square root of a negative number is not a real number, so the two roots are complex. They come in a conjugate pair: one is real_part + imaginary_part·i and the other is real_part − imaginary_part·i. The real part is −b / 2a and the imaginary part is √(−D) / 2a.

Yes, but only when the discriminant is exactly zero. In that case, the formula collapses to a single value, x = −b / 2a, which is sometimes called a repeated or double root because it counts twice in the algebra.

If a = 0, the x² term disappears and the equation becomes bx + c = 0, which is linear, not quadratic. The quadratic formula divides by 2a, so a = 0 would also mean dividing by zero. The calculator flags this case and asks you to enter a nonzero a.

Yes. The inputs accept decimals, fractions written as decimals, and negative numbers. Very large or unusual values are handled gracefully; the calculator rounds the displayed roots for readability while computing internally at full floating-point precision.

The difference is that the quadratic formula is a universal method that solves any quadratic equation, whereas factoring is a shortcut that only works easily when the roots are integers. If a quadratic expression cannot be factored using simple numbers, the quadratic formula must be used to find the solutions.

The variables a, b, and c represent the numerical coefficients of the terms in a standard quadratic equation. Specifically, a is the coefficient of the squared term (x²), b is the coefficient of the linear term (x), and c is the constant number.

You use completing the square primarily when solving quadratic equations that are not easily factorable or when rewriting a standard form quadratic function into vertex form to find its vertex. While the quadratic formula is generally faster for finding roots, completing the square is essential for understanding the algebraic origin of the formula itself.

No, a quadratic equation always has exactly two solutions, though they may be real and distinct, real and identical, or complex conjugate numbers. In real-world problems where only real-number solutions are physically possible, a negative discriminant is often interpreted as having no real solutions.

The term −b / 2a in the quadratic formula gives the x-coordinate of the parabola's vertex (which is also the axis of symmetry). The roots of the equation are spaced symmetrically at a distance of √(b² − 4ac) / 2a to the left and right of this vertex line.