Quadratic Equation Solver
Free quadratic equation solver. Find real and complex roots of ax² + bx + c = 0 with the quadratic formula and discriminant. Step-by-step.
What is a quadratic equation solver?
A quadratic solver finds the values of x that satisfy ax² + bx + c = 0 using the quadratic formula. It returns real roots, a double root, or complex roots depending on the discriminant — the part that decides how many solutions exist.
How this calculator works
- Enter the coefficients a, b, and c from your equation in standard form.
- It computes the discriminant (b² − 4ac) and displays it alongside the roots.
- Positive discriminant → two real roots; zero → one double root; negative → two complex roots.
- If a is 0, it falls back to solving the linear equation bx + c = 0.
How to use the result
- Rewrite your equation so everything is on one side and equal to zero before entering coefficients.
- Use the discriminant to check your graph: negative means the parabola never crosses the x-axis.
- If roots are complex, real-world problems usually still have a real interpretation — revisit the model that produced the equation.
- For physics or geometry problems, verify your coefficients have the right signs — a missing minus sign flips the roots.
What does the discriminant tell me?
The discriminant b² − 4ac predicts the nature of the roots without solving. If it’s positive you get two distinct real solutions (the parabola crosses the x-axis twice), zero means one repeated root (it just touches), and negative means complex roots (it never touches the axis). It’s the quickest check on whether your equation has real answers at all.
Common mistakes
- Forgetting to move everything to one side. x² + 3x = 10 must become x² + 3x − 10 = 0 first.
- Sign errors on b and c. The formula uses −b, so a positive b flips the sign in the numerator.
- Assuming a is never 0. If a = 0 it’s not quadratic — the calculator handles the linear case.
Quadratic equations are the backbone of the Pythagorean theorem’s geometry — verify a right-triangle side with the Pythagorean calculator if your problem is spatial. For quick arithmetic checks while rearranging equations, the basic calculator is handy.