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Quadratic Equation Calculator
Last updated: May 31, 2026
Written by Blake Boege
A quadratic equation calculator is an algebraic solver designed to find the roots of any second-degree polynomial equation. Given the coefficients of the quadratic term, linear term, and constant, the tool computes the discriminant to determine the number and type of roots (real, repeated, or complex). It then applies the quadratic formula to solve for x. This calculator is widely used in algebra courses, physics simulations, and engineering to solve projectile, optimization, and parabolic problems.
Solve second-degree polynomial equations of the form ax² + bx + c = 0. Get the discriminant, root classification, and full step-by-step derivations.
Quick Answer
Solve second-degree quadratic equations of the form ax² + bx + c = 0. Enter coefficients a, b, and c to see real or complex roots.
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. Identify a, b, and c. a = 1, b = -5, c = 6.
- 2. Calculate the discriminant. D = -5² − 4·1·6 = 25 − 24 = 1.
- 3. Plug into the formula. x = (−(-5) ± √1) / 2·1
- 4. Simplify. x₁ = 3 and x₂ = 2.
Roots
3, 2
Two distinct real roots from D = 1
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.
Examples
x² − 5x + 6 = 0 [Factoring case]
x = 2, x = 3
x² + 4x + 4 = 0 [Repeated root]
x = −2
x² + 2x + 5 = 0 [Complex roots]
x = −1 ± 2i
How it works
For any quadratic equation in standard form ax² + bx + c = 0, the solutions are calculated using the quadratic formula:
Quadratic Formula · x = (−b ± √(b² − 4ac)) / 2a
Discriminant · D = b² − 4ac
The value of the discriminant determines the solution types:
- D > 0: Two distinct real roots
- D = 0: One repeated real root (double root)
- D < 0: Two complex conjugate roots
Methods for solving quadratic equations
Depending on the coefficients, algebraic equations can be solved using several approaches:
- Factoring: Rewriting the quadratic as product factors (e.g., (x − 2)(x − 3) = 0). This is fastest when the roots are integers.
- Quadratic Formula: The universal algebraic method that solves any quadratic equation, regardless of whether the roots are fractional, irrational, or complex.
- Completing the Square: Rearranging the terms to form a perfect square trinomial, which is the algebraic method used to derive the quadratic formula itself.
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Frequently asked questions
A quadratic equation is a second-degree polynomial equation in a single variable, expressed in standard form as ax² + bx + c = 0, where a, b, and c are constant coefficients and a is not equal to zero.
Quadratic equations can be solved using factoring, completing the square, graphing, or by applying the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. The formula works for all quadratic equations.
The discriminant is the expression under the radical in the quadratic formula: D = b² − 4ac. It determines the number and nature of the solutions. D > 0 yields two distinct real solutions; D = 0 yields one repeated real solution; D < 0 yields two complex conjugate solutions.
Yes. When the discriminant (b² − 4ac) is negative, you must take the square root of a negative number. This yields two complex solutions containing the imaginary unit i (where i = √−1).
If a = 0, the x² term is eliminated, transforming the equation into a first-degree linear equation (bx + c = 0). The quadratic formula also divides by 2a, which would result in division by zero if a were zero.
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