Solve any quadratic equation of the form ax² + bx + c = 0. Find both roots, discriminant, and vertex with step-by-step solution.
This tool solves quadratic equations of the form ax² + bx + c = 0, finding the values of x that satisfy the equation. It's a fundamental algebra tool used in math courses, physics problems (like projectile motion), and engineering calculations.
The quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant (b² − 4ac) determines the nature of the solutions: positive means two real solutions, zero means one repeated real solution, and negative means two complex (imaginary) solutions.
Enter the coefficients a, b, and c from your quadratic equation, and the calculator returns the value(s) of x that solve it.
Example: For x² − 5x + 6 = 0 (a=1, b=−5, c=6), the discriminant is 25 − 24 = 1, giving two solutions: x = 3 and x = 2.
What does the discriminant tell you? The discriminant (b² − 4ac) reveals how many real solutions exist without fully solving the equation: positive means two distinct real roots, zero means exactly one repeated root, and negative means no real roots (only complex ones).
What if the equation has no real solutions? This happens when the discriminant is negative, meaning the solutions involve imaginary numbers (using i, where i² = −1) — the parabola representing the equation doesn't cross the x-axis in this case.
Why is it called "quadratic"? The term comes from "quadratus," Latin for square, referring to the x² term (x squared) that defines this type of equation.
What real-world situations use quadratic equations? Quadratic equations model projectile motion (like a thrown ball's trajectory), profit optimization in business, and any relationship where one quantity depends on the square of another.
Can quadratic equations be solved without the quadratic formula? Yes, alternative methods include factoring (when the equation factors neatly), completing the square, or graphing to find where the parabola crosses the x-axis — the quadratic formula works universally when these other methods are difficult to apply.