What Is a Quadratic Formula Calculator?
A quadratic formula calculator solves equations of the form ax² + bx + c = 0, returning both roots (solutions) along with the discriminant that reveals what kind of solutions to expect. Quadratics appear everywhere in algebra, physics, engineering, and finance — modeling projectile motion, areas, profit curves, and more. Instead of factoring by hand or completing the square, you enter the coefficients a, b, and c and get exact answers instantly, including real and complex roots.
How to Use the Quadratic Formula Calculator
- Enter coefficient a — the number in front of x² (cannot be zero).
- Enter coefficient b — the number in front of x.
- Enter coefficient c — the constant term.
- Calculate — see both roots, the discriminant, and the worked solution.
The Quadratic Formula
The roots of ax² + bx + c = 0 are given by:
x = (−b ± √(b² − 4ac)) ÷ (2a)
The two solutions come from the plus and minus options on the square root. The expression under the root, b² − 4ac, is called the discriminant, and it determines the nature of the roots before you finish solving.
What the Discriminant Tells You
| Discriminant (b² − 4ac) | Roots |
|---|---|
| Positive | Two distinct real roots |
| Zero | One repeated real root |
| Negative | Two complex (imaginary) roots |
Worked Example
Solve x² − 5x + 6 = 0, so a = 1, b = −5, c = 6. The discriminant is (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, so there are two real roots. Applying the formula: x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, giving x = 3 and x = 2.
Other Ways to Solve Quadratics
- Factoring: rewrite as (x − r₁)(x − r₂) = 0 when the roots are simple integers.
- Completing the square: reshape into a perfect-square trinomial; this is how the quadratic formula is derived.
- Graphing: the roots are where the parabola crosses the x-axis.
- The quadratic formula: always works, even when factoring is difficult or roots are irrational or complex.
Frequently Asked Questions
What is the quadratic formula?
It is x = (−b ± √(b² − 4ac)) ÷ (2a), the universal method for solving any equation of the form ax² + bx + c = 0.
What is the discriminant?
The discriminant is b² − 4ac. A positive value gives two real roots, zero gives one repeated root, and a negative value gives two complex roots.
Can a quadratic have no real solutions?
Yes. When the discriminant is negative, the equation has no real roots — instead it has two complex (imaginary) solutions, which this calculator can also display.
Why must a not equal zero?
If a is zero, the x² term disappears and the equation becomes linear (bx + c = 0), not quadratic, so the quadratic formula no longer applies.
Is this quadratic calculator free?
Yes — it is completely free, requires no signup, and shows the full step-by-step solution.