Quadratic equation solver
Enter a quadratic equation to find real or complex roots, then review the discriminant, vertex, axis of symmetry, graph, and verification.
Try an example
A quadratic equation can be written as ax² + bx + c = 0 with a ≠ 0. It can have two real roots, one repeated real root, or a conjugate pair of complex roots.
The calculator accepts x², x^2, x*x, and products that expand to degree two, such as (x - 2)(x - 3) = 0.
When real roots are obvious integers, the solver factors the quadratic and applies the zero-product property. Otherwise it calculates the discriminant and uses the quadratic formula.
The discriminant determines the root type. A negative discriminant now produces the complex roots, while the calculator also reports the parabola’s vertex and axis of symmetry.
Type or paste an equation, solve it, inspect the transformations, and use verification and the graph instead of relying on the final value alone.
x² - 5x + 6 = 0x² + 4x + 4 = 02x² - 8 = 0x² - 2 = 0x² + 1 = 0Follow the same steps the calculator uses for three equations in this category.
x = 2, x = 3
Factor the quadratic expression
x^2-5x+6=0 → (x-2)(x-3)=0
Use the zero-product property to solve each factor
(x-2)(x-3)=0 → x=2 or x=3
Verified
x = -2
Factor the quadratic expression
x^2+4x+4=0 → (x+2)^2=0
Use the zero-product property to solve each factor
(x+2)^2=0 → x=-2
Verified
x = -2, x = 2
Factor the quadratic expression
2x^2-8=0 → 2(x+2)(x-2)=0
Use the zero-product property to solve each factor
2(x+2)(x-2)=0 → x=-2 or x=2
Verified
Substitute each answer into the original equation and compare both sides.
Cubic and higher-degree polynomials, logarithmic, exponential, trigonometric, nested-radical, and multiple-absolute-value equations remain outside the supported solver scope.