Quadratic Equation Solver

Solve quadratic equations with the discriminant and quadratic formula shown step by step, including exact radical and complex roots.

Your result will appear here

Fill in the fields and press Solve, or pick one of the examples.

About the Quadratic Equation Solver

Solves $ax^2 + bx + c = 0$ with the quadratic formula and shows the discriminant, which decides the nature of the roots: two real roots when it is positive, one repeated root when it is zero, and a pair of complex conjugates when it is negative.

With whole-number or fractional coefficients the roots are exact: fractions when the discriminant is a perfect square, simplified radicals such as $\frac{1 + \sqrt{5}}{2}$ otherwise.

How to use it

Enter the equation in any arrangement, e.g. x^2 = 5x - 6.
Press Solve.
Read the coefficients, discriminant and roots in the working.

Example

x^2 - 5x + 6 = 0

→ x₁ = 2, x₂ = 3 (discriminant 1)

Formulas used

x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

The quadratic formula.

\Delta = b^{2} - 4ac

The discriminant. Its sign gives the number of real roots.

x_1 + x_2 = -\frac{b}{a},\quad x_1 x_2 = \frac{c}{a}

Vieta’s formulas link the roots to the coefficients.

Questions and answers

What if the discriminant is negative?

There are no real solutions. The solver shows the two complex conjugate roots; choose “Real roots only” to report “no real solutions” instead.

Completing the SquareRewrite a quadratic in vertex form a(x − h)² + k with every step, and read off the vertex, axis of symmetry and extreme value.Expression FactorizerFactor polynomials and algebraic expressions over the rationals, with the result checked against the original expression.Polynomial Equation SolverFind all real and complex roots of polynomial equations: exact rational roots first, then numerical roots for the rest.Graphing CalculatorPlot and compare functions, implicit equations and inequalities with zoom, pan, coordinate tracking and PNG export.

About the Quadratic Equation Solver

How to use it

Formulas used

Questions and answers

Related tools

About the Quadratic Equation Solver

How to use it

Formulas used

Questions and answers