D > 0
There are two distinct real roots. The parabola crosses the x-axis at two points.
Math
Quadratic equation calculator
Calculate real or complex roots with the discriminant, vertex, exact form and quadratic-formula steps.
Coefficients
Equation
ax² + bx + c = 0
Examples
Quadratic-formula steps
x = (-b ± sqrt(D)) / (2a)
Substitute the coefficients to view the formula.
- Complete the values to view the steps.
Publicidad
How to read the result
The discriminant connects the roots with the shape of the parabola.
D = 0
There is one double real root. The parabola touches the x-axis at its vertex.
D < 0
There are no real roots. The solutions appear as complex numbers.
Quadratic formula
For an equation ax² + bx + c = 0, the discriminant is D = b² - 4ac, and the solutions are obtained with x = (-b ± sqrt(D)) / (2a).
Frequently asked questions
What is the discriminant?
The discriminant is D = b² - 4ac. Its sign indicates whether the equation has two real roots, one double root or complex roots.
What happens when the discriminant is negative?
When the discriminant is negative, the equation has no real roots. The calculator shows two complex roots with real and imaginary parts.
Why can't a be zero?
If a = 0, the expression is no longer quadratic and becomes a linear equation of the form bx + c = 0.
Does the calculator show complex roots?
Yes. When D < 0, it shows solutions as real part ± imaginary part i.
What does a double root mean?
A double root occurs when D = 0. On the graph, the parabola touches the x-axis at one point.
What does the parabola's vertex represent?
The vertex is the parabola's maximum or minimum point. It is calculated with xv = -b/(2a) and yv = a*xv² + b*xv + c.