Two distinct real roots
When the discriminant is greater than zero, the parabola crosses the x-axis at two different points.
x² - 5x + 6 = 0
Roots: x = 2 and x = 3
Quadratic Formula Calculator
Use this quadratic formula calculator to solve any quadratic equation in the form ax² + bx + c = 0. Get step-by-step solutions and find the roots instantly.
Quadratic Formula Calculator
Enter the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0, then click Calculate to find the roots.
$ax^2 + bx + c = 0$
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable x, where a, b, and c are constants, and a ≠ 0. The standard form is ax² + bx + c = 0.
Quadratic equations can have either one, two, or no real solutions, depending on the value of the discriminant (b² - 4ac). When graphed, a quadratic equation forms a parabola.
The Quadratic Formula
The quadratic formula provides a way to solve any quadratic equation:
$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
The expression under the square root (b² - 4ac) is called the discriminant (Δ) and determines the nature of the roots:
$\Delta = b^2 - 4ac$
If Δ > 0, there are two distinct real roots. If Δ = 0, there is one repeated real root. If Δ < 0, there are two complex conjugate roots.
Applications of Quadratic Equations
Physics
Used in projectile motion, free-falling objects, and many other physical phenomena that involve acceleration.
Engineering
Applied in structural analysis, electrical circuits, and optimization problems in various engineering disciplines.
Finance
Used in calculating compound interest, asset pricing models, and option pricing in financial mathematics.
Computer Science
Applied in algorithm analysis, computer graphics for parabolic curves, and machine learning models.
How to Use This Calculator
- Enter the coefficient a (the coefficient of x²) in the first input field.
- Enter the coefficient b (the coefficient of x) in the second input field.
- Enter the coefficient c (the constant term) in the third input field.
- Click 'Calculate' to solve the equation and see the step-by-step solution.
Coefficient a cannot be zero. If the discriminant is positive you get two real roots, if it is zero you get one repeated root, and if it is negative you get complex roots.
Quadratic Equation Examples
One repeated real root
When the discriminant equals zero, the parabola touches the x-axis at exactly one point.
x² - 4x + 4 = 0
Root: x = 2
Complex roots
When the discriminant is negative, the equation has no real x-intercepts but still has two complex solutions.
x² + 2x + 5 = 0
Roots: x = -1 ± 2i
Quadratic Formula FAQ
What does the discriminant tell me?
The discriminant b² - 4ac tells you whether the quadratic has two real roots, one repeated root, or two complex roots.
Why can't coefficient a be zero?
If a equals zero, the x² term disappears and the equation is no longer quadratic.
When should I use the quadratic formula instead of factoring?
Use the quadratic formula when factoring is difficult, unreliable, or impossible to do quickly by inspection.
Can the calculator return complex roots?
Yes. When the discriminant is negative, the calculator shows the complex conjugate roots.