Quadratic Formula Calculator Online Free
Quadratic Formula Calculator
Quadratic Equation Solver
Example: For x² + x + 1/4 = 0, enter a=1, b=1, c=1/4
(1)x² + (1)x + (1/4) = 0
Cannot be zero
Linear coefficient
Constant term
The quadratic formula solves any equation of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to instantly find the roots (solutions) of the equation, along with whether they are real or complex. This is one of the most useful tools in algebra, with applications ranging from physics to finance.
The Quadratic Formula
For any quadratic equation ax² + bx + c = 0, the solutions are found using the quadratic formula. The discriminant (b² - 4ac) tells you the nature of the solutions before you calculate them.
x = (-b ± √(b² - 4ac)) / (2a) Discriminant (D) = b² - 4ac: • D > 0: Two distinct real roots • D = 0: One repeated real root • D < 0: Two complex (imaginary) roots
The ± means you get two solutions: one with + and one with -.
Understanding the Discriminant
The discriminant tells you the relationship between the parabola and the x-axis. When D > 0, the parabola crosses the x-axis twice. When D = 0, it touches it exactly once (the vertex is on the x-axis). When D < 0, it never crosses the x-axis, and the solutions involve the imaginary number i = √(-1).
| Discriminant | Number of Real Roots | Parabola Behavior |
|---|---|---|
| D > 0 | Two distinct real roots | Crosses x-axis twice |
| D = 0 | One repeated root | Touches x-axis at vertex |
| D < 0 | No real roots | Does not cross x-axis |
Frequently Asked Questions
When should I use the quadratic formula vs factoring?⌄
Factoring is faster when the equation factors neatly (like x² - 5x + 6 = (x-2)(x-3)). Use the quadratic formula when factoring is not obvious, when coefficients are not integers, or when you need exact answers including irrational numbers. The quadratic formula always works regardless of whether the equation is factorable.
What does it mean when a quadratic has no real solutions?⌄
It means the parabola y = ax² + bx + c does not intersect the x-axis. The solutions are complex numbers involving i (the square root of -1). In practical contexts (like finding dimensions or time), complex solutions typically mean the original scenario has no real-world solution with those parameters.
What are the real-world applications of quadratic equations?⌄
Projectile motion (height of a thrown ball over time), area optimization problems (maximum fenced area given a fixed perimeter), profit maximization in business (revenue minus cost as a function of quantity), and many physics and engineering applications that involve squared variables.
What is completing the square?⌄
Completing the square is an alternative method for solving quadratics, where you algebraically rearrange the equation into the form (x + p)² = q, then take the square root of both sides. The quadratic formula is actually derived by completing the square on the general form ax² + bx + c = 0.