The Quadratic Formula
Every second-degree polynomial equation in a single variable can be solved by evaluating the universal quadratic formula:
Universal Quadratic Formula
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
Discriminant Definition
$$\Delta = b^2 - 4ac$$
Example ($$1x^2 - 3x + 2 = 0$$):
$$\Delta = (-3)^2 - 4(1)(2) = 9 - 8 = \mathbf{1}$$
$$x = \frac{-(-3) \pm \sqrt{1}}{2(1)} = \frac{3 \pm 1}{2} \implies \mathbf{x_1 = 2, x_2 = 1}$$
Discriminant Classification & Root Behavior
The sign of the discriminant ($$\Delta = b^2 - 4ac$$) determines the number and numerical character of the solutions:
| Discriminant Value | Nature of Roots | Graphical Interpretation | Example Equation |
|---|---|---|---|
| $$\Delta > 0$$ | Two distinct real roots | Parabola crosses x-axis at two points | $$x^2 - 3x + 2 = 0$$ |
| $$\Delta = 0$$ | One repeated real root | Parabola vertex touches x-axis at one point | $$x^2 - 4x + 4 = 0$$ |
| $$\Delta < 0$$ | Two complex conjugate roots ($$a \pm bi$$) | Parabola does not intersect x-axis | $$x^2 + 2x + 5 = 0$$ |
What the Discriminant Tells You Before Solving
The discriminant ($$\Delta = b^2 - 4ac$$) reveals essential geometric information prior to completing the equation. A positive result indicates two distinct real roots. A discriminant of zero signifies a single repeated root where the parabola's vertex rests on the axis. A negative value yields complex conjugate pairs because finding real solutions requires taking square roots of negative numbers.
How to Use This Calculator
- Enter values for coefficients a, b, and c into their respective input fields.
- Ensure coefficient a is non-zero (since $$a = 0$$ produces a linear equation).
- The roots (real or complex) are calculated dynamically in real time.
Frequently Asked Questions
What does the discriminant tell me?
A positive discriminant gives two distinct real roots, zero gives one repeated real root, and a negative discriminant gives two complex conjugate roots.
Can this solve equations with complex roots?
Yes, when the discriminant is negative, the calculator displays both complex conjugate roots in standard a ± bi form.
What if 'a' is zero?
If a = 0, the equation becomes linear (bx + c = 0) rather than quadratic. Enter a non-zero number for coefficient a.
Why are there normally two solutions?
Quadratic equations plot a parabola on a cartesian plane, which typically intersects the x-axis at up to two distinct points.