Quadratic Formula Calculator
Solve quadratic equations with the quadratic formula
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Frequently Asked Questions
What is the quadratic formula?
For ax^2 + bx + c = 0: x = (-b ± sqrt(b^2 - 4ac)) / (2a). The discriminant b^2 - 4ac determines the type of roots: positive = two distinct real roots; zero = one repeated real root; negative = two complex conjugate roots.
What is the discriminant?
The discriminant is b^2 - 4ac. If > 0: two distinct real roots (parabola crosses x-axis twice). If = 0: one repeated real root (parabola is tangent to x-axis). If < 0: no real roots, two complex conjugate roots (parabola does not cross x-axis).
How do I factor a quadratic equation?
To factor x^2 - 5x + 6 = 0: find two numbers that multiply to c (6) and add to b (-5). Those are -2 and -3: (x-2)(x-3) = 0, so x = 2 or x = 3. When factoring is difficult, use the quadratic formula — it always works.
What is the vertex of a parabola?
The vertex is the minimum or maximum point of the parabola y = ax^2 + bx + c. Vertex x = -b/(2a). Vertex y = c - b^2/(4a). If a > 0, it is a minimum (parabola opens up). If a < 0, it is a maximum (parabola opens down).
Can I solve a quadratic by completing the square?
Yes. For x^2 + bx + c = 0: (1) Move c: x^2 + bx = -c. (2) Add (b/2)^2 to both sides: x^2 + bx + (b/2)^2 = -c + (b/2)^2. (3) Factor left side: (x + b/2)^2 = discriminant/4. (4) Solve: x = -b/2 ± sqrt(discriminant/2). This is essentially how the quadratic formula is derived.