Enter coefficients a, b, and c to solve ax² + bx + c = 0 instantly. See roots, vertex, discriminant, and a graph.
A quadratic equation is a polynomial equation of the second degree, written in the standard form ax² + bx + c = 0, where a is not equal to zero. The solutions (called roots or zeros) are the values of x that make the equation true. A quadratic can have two real solutions, one repeated real solution, or two complex (imaginary) solutions, depending on the discriminant.
Enter the coefficients a, b, and c from your equation in standard form. The calculator applies the quadratic formula to find both roots, shows the discriminant value, identifies whether the roots are real and distinct, real and repeated, or complex, and displays the work step by step. It also shows the vertex and axis of symmetry of the parabola.
The discriminant (b² - 4ac) tells you the nature of the roots without solving the equation. If the discriminant is positive, there are two distinct real roots. If it equals zero, there is one repeated real root (the parabola just touches the x-axis). If it is negative, there are two complex conjugate roots (the parabola does not cross the x-axis).
Quadratic equations model parabolic curves that appear throughout physics and engineering. Projectile motion follows a parabola: the height of a ball thrown upward equals h = -16t^2 + vt + h0 (in feet, where v is initial velocity and h0 is initial height). Solving this quadratic tells you when the ball hits the ground. Bridge cables form parabolas under uniform load. Satellite dish and headlight reflectors use parabolic shapes to focus signals and light. In business, profit functions are often quadratic (revenue grows linearly with price up to a point, then demand drops). The discriminant (b^2 - 4ac) determines the solution type: positive means two real roots, zero means one repeated root (the vertex touches the x-axis), and negative means no real roots (the parabola does not cross the x-axis).