ax² + bx + c = 0
x = (-b ± √(b² - 4ac)) / 2a
This formula gives the solutions (roots) of any quadratic equation in the form ax² + bx + c = 0.
This calculator provides estimates only. Verify manually for critical calculations. Results are rounded for display but calculations use full precision.
A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the standard form ax² + bx + c = 0, where x represents an unknown variable and a, b, and c are constants with a ≠ 0. The term "quadratic" comes from "quadratus," the Latin word for square, because the variable is squared (raised to the second power). Quadratic equations are fundamental in mathematics and appear frequently in physics, engineering, economics, and many other fields.
The solutions to a quadratic equation, also called roots or zeros, are the values of x that make the equation true. These roots represent the points where the parabola (the graph of a quadratic function) crosses the x-axis. A quadratic equation can have two distinct real roots, one repeated real root, or two complex conjugate roots, depending on the value of the discriminant.
The discriminant, denoted as D or Δ, is the expression b² - 4ac found under the square root in the quadratic formula. It plays a crucial role in determining the nature and number of roots without actually solving the equation. The discriminant acts as a "discriminator" that tells us whether the roots are real or complex, and whether they are distinct or equal.
When D > 0 (Positive Discriminant)
The equation has two distinct real roots. The parabola crosses the x-axis at two different points. This is the most common case in practical applications. For example, x² - 5x + 6 = 0 has D = 1, giving roots x = 2 and x = 3.
When D = 0 (Zero Discriminant)
The equation has exactly one real root (or two equal real roots). The parabola touches the x-axis at exactly one point - the vertex. This root is called a "repeated" or "double" root. For example, x² - 6x + 9 = 0 has D = 0, giving the repeated root x = 3.
When D < 0 (Negative Discriminant)
The equation has two complex conjugate roots. The parabola does not cross the x-axis at all. The roots involve the imaginary unit i = √(-1). For example, x² + x + 1 = 0 has D = -3, giving roots x = (-1 + i√3)/2 and x = (-1 - i√3)/2.
While the quadratic formula always works, there are several methods for solving quadratic equations, each with its own advantages depending on the specific equation.
Factoring: When the quadratic can be easily factored, this method is often the quickest. For example, x² - 5x + 6 = (x - 2)(x - 3) = 0 immediately gives x = 2 or x = 3. However, not all quadratics factor nicely with integer coefficients.
Completing the Square: This method transforms the equation into a perfect square trinomial. It's particularly useful for deriving the quadratic formula and for problems involving circles and parabolas. The technique involves adding and subtracting (b/2a)² to create a perfect square.
Quadratic Formula: The most universal method, x = (-b ± √(b² - 4ac)) / 2a works for any quadratic equation. It's especially useful when factoring is difficult or when dealing with coefficients that aren't integers. This calculator uses this method for its reliability and precision.
Graphical Method: By plotting y = ax² + bx + c and finding where it crosses the x-axis, you can visually identify the roots. While less precise than algebraic methods, it provides valuable intuition about the nature and approximate values of the roots.
Quadratic equations appear in countless real-world situations. In physics, projectile motion follows a parabolic path, and the height of an object thrown upward can be modeled by h(t) = -½gt² + v₀t + h₀, where finding when h(t) = 0 tells us when the object hits the ground. Engineers use quadratic equations to design parabolic antennas and bridges, calculate stress distributions, and optimize electrical circuits.
In economics and business, quadratic equations model profit optimization, where the profit function P(x) = -ax² + bx - c represents revenue minus costs. Finding the vertex gives the maximum profit point. In finance, compound interest over time can create quadratic relationships. Even in everyday situations like determining the dimensions of a garden with a fixed perimeter to maximize area, or calculating braking distances, quadratic equations provide the mathematical framework for finding solutions.