What a quadratic equation is
A quadratic equation, or second-degree equation, has the form ax² + bx + c = 0, where a, b and c are known numbers and a cannot be zero. If a were zero, the x² term would vanish and you would be left with an ordinary linear equation. Solving it means finding the values of x that make the equality true; those values are called the roots or solutions.
These equations show up everywhere: the flight of a thrown ball, the area of a rectangular plot when you know its perimeter, or the break-even point of a business. That is why solving them quickly and without sign mistakes matters, and it is exactly what this calculator does right in your browser, without sending any data.
How to use the calculator
- Type coefficient a (remember: it must not be zero).
- Type b and c. If you leave either one blank it is treated as 0.
- Read the roots, the discriminant and the vertex of the parabola instantly. There is no “calculate” button to press.
- Press Copy result to drop the summary into your notes.
The quadratic formula
The roots come from the general formula:
x = (−b ± √D) / 2a, where D = b² − 4ac
The number D is called the discriminant, and it decides how many roots there are and of what kind without solving the whole equation. The vertex of the parabola sits at x = −b / 2a, and its height is y = c − b² / 4a.
| Discriminant | Nature of the roots | Example |
|---|---|---|
D greater than 0 | Two distinct real roots | x² − 5x + 6 = 0 |
D equal to 0 | One real double root | x² + 2x + 1 = 0 |
D less than 0 | Two complex conjugate roots | x² + 1 = 0 |
When D is negative there is no solution among the real numbers, but there is among the complex ones: the roots are real part ± imaginary part · i, where i is the square root of −1.
Worked example
Let’s solve x² − 5x + 6 = 0, so a = 1, b = −5 and c = 6.
- Discriminant: D = (−5)² − 4·1·6 = 25 − 24 = 1.
- Since
Dis greater than 0, there are two real roots. - Square root of D: √1 = 1.
- x₁ = (5 + 1) / 2 = 6 / 2 = 3.
- x₂ = (5 − 1) / 2 = 4 / 2 = 2.
- Vertex: x = 5 / 2 = 2.5, y = 6 − 25 / 4 = −0.25.
Check: 3 × 2 = 6 (which is c) and 3 + 2 = 5 (which is −b), exactly as the relationship between roots and coefficients predicts. A second quick case: 2x² − 4x − 6 = 0 gives D = 16 + 48 = 64, with roots 3 and −1 and a vertex at (1, −8).
Frequently asked questions
What happens if a = 0?
It is no longer a quadratic equation. With a = 0 the expression becomes bx + c = 0, a linear equation whose solution is x = −c / b. The calculator warns you and asks for a value of a other than zero.
Why am I getting complex roots?
Because the discriminant is negative: the parabola never crosses the horizontal axis, so there are no real solutions. The answers are written as number ± number · i. They are perfectly valid and widely used in physics and engineering.
What is the discriminant for?
It tells you at a glance how many roots there are and of what type, without solving the full equation. Positive means two real roots; zero means one double root; negative means two complex roots. It also says whether the parabola touches, crosses, or misses the x-axis.
What does the vertex represent?
It is the highest or lowest point of the parabola y = ax² + bx + c. If a is positive the parabola opens upward and the vertex is the minimum; if a is negative it opens downward and the vertex is the maximum. Its x-coordinate, −b / 2a, is also the axis of symmetry.
Does the calculator round?
Yes, it shows up to 4 decimals so the number stays readable, and it avoids the annoying “−0”. Internally it computes at your device’s full precision, so you can trust the result for homework and quick checks.