Solve quadratic equations of the form ax² + bx + c = 0. Supports fractions (e.g. 1/4 or 3/4) and displays detailed calculations step-by-step.
The Quadratic Equation Calculator is an algebraic tool designed to solve quadratic equations and find their real or complex roots. A quadratic equation is a second-order polynomial equation in a single variable, commonly written in the standard form: ax² + bx + c = 0, where x represents the unknown variable, and a, b, and c are numerical coefficients (with a not equal to zero). Quadratic equations are a cornerstone of algebra, modeling curved paths, projectile motions, and geometric areas.
To find the roots (solutions) of a quadratic equation, the calculator utilizes the quadratic formula: x = [−b ± √(b² − 4ac)] / 2a. The term under the square root, b² − 4ac, is called the "discriminant" (commonly represented by the symbol Δ). The discriminant determines the nature and number of the roots: 1. If the discriminant is positive (Δ > 0), the equation has two distinct real roots. 2. If the discriminant is zero (Δ = 0), the equation has exactly one real root (a repeated root). 3. If the discriminant is negative (Δ < 0), the equation has two complex conjugate roots containing the imaginary unit i (where i = √−1).
Quadratic equations are used in physics to model projectile motion under gravity, calculating the trajectory and landing time of objects. In geometry, they determine the dimensions of shapes given their area. In economics, quadratic functions model profit maximization curves, helping businesses identify optimal pricing. By providing step-by-step solutions for real and complex roots, the quadratic calculator simplifies algebraic homework, structural physics planning, and geometric proofs.
Understanding the Quadratic Formula
In algebra, a **quadratic equation** is any polynomial equation of the second degree. It represents a parabola when graphed and takes the standard form:
Here, x is the unknown variable, a is the quadratic coefficient (which cannot be zero), b is the linear coefficient, and c is the constant. The values of x that satisfy this equation are called the roots or **x-intercepts** where the parabola crosses the horizontal x-axis.
Derivation of the Quadratic Formula (Completing the Square)
The quadratic formula is derived by solving the general equation ax² + bx + c = 0 for x using the technique of completing the square:
Properties of a Quadratic Parabola
- Axis of Symmetry: The vertical line
x = -b / (2a)that cuts the parabola exactly in half. - Vertex: The turning point of the parabola, represented by coordinates
(h, k)whereh = -b / (2a). - Direction: If
a > 0, the parabola opens upwards (minimum vertex). Ifa < 0, it opens downwards (maximum vertex).
Key entities: Quadratic Equation + Roots + Completing the Square + Discriminant.
How it Works & Formula
Computes the roots using the quadratic formula. Calculates the discriminant to determine real vs. complex roots, along with parabola coordinates.
Practical Examples
For x² + x + 1/4 = 0, the discriminant is 0, yielding a single root x = -0.5.
For x² - 5x + 6 = 0, roots are x = 2 and x = 3.
For x² + 2x + 5 = 0, roots are x = -1 ± 2i.
Frequently Asked Questions
What is a quadratic equation?
A quadratic equation is a second-degree polynomial equation of the form ax² + bx + c = 0, where x is the variable, and a, b, and c are constant coefficients (with a ≠ 0).
What does the discriminant (b² - 4ac) tell us?
The discriminant determines the nature of the roots. If positive, there are 2 real roots. If zero, there is 1 double real root. If negative, there are 2 complex (imaginary) roots.
Why cannot "a" equal zero?
If a = 0, the x² term vanishes, transforming the equation into bx + c = 0, which is a linear equation (first-degree polynomial).