Quadratic Equation Calculator

Find the roots of quadratic equations instantly and review the step-by-step discriminant working.

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Enter the coefficients for a, b, and c to solve the equation. The workspace handles positive, negative, and decimal values.

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ax²

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A Guide to Solving Quadratic Equations

In algebra, a quadratic equation is any polynomial equation of the second degree. This means it contains at least one term where a variable is squared (raised to the power of two). The standard form of this equation is written as ax² + bx + c = 0, where a, b, and c represent numerical coefficients. Finding the values of x that make the equation true is a core task in advanced math and physics.

The Quadratic Formula

To solve these equations, mathematicians use the quadratic formula. This formula uses the coefficients a, b, and c to calculate the roots. Because the formula includes a plus-or-minus symbol, it typically yields two separate solutions, representing the two points where the curve of the equation crosses the horizontal axis of a graph.

The formula requires working with squared terms, which you can explore with our number powers calculator, and square roots, which you can resolve using our finding radical roots tool. For linear equations that form straight lines, you would use a calculating linear slopes tool instead.

Understanding the Discriminant

The value under the radical sign in the formula (b² - 4ac) is called the discriminant. It tells you the nature of the roots before you solve the whole equation:

Positive Discriminant: The equation has two distinct real number solutions, crossing the axis at two separate points.
Zero Discriminant: The equation has exactly one real number solution, touching the axis at only one point.
Negative Discriminant: The equation has no real solutions, resulting in two complex numbers containing imaginary values.

Real-World Quadratic Scenarios

Quadratic equations are heavily used to model parabolic motion in physics. When you throw a ball, launch a rocket, or jump off a diving board, the path you travel forms a parabola. Solving the quadratic equation tells you exactly when the object will hit the ground.

These equations are also used in business to optimize prices for maximum revenue and in geometry to calculate dimensions. For example, if you are designing a rectangular garden bed, you might need to solve a quadratic equation to find the border lengths, which you can check using our solving triangle values tool or a standard math tool.

Example of Projectile Trajectory

Imagine a toy rocket is launched straight up into the air, and its height over time is represented by the quadratic equation -16t² + 64t = 0, where t is time in seconds.

To find when the rocket returns to the ground, we solve for t. The coefficients are a = -16, b = 64, and c = 0. Plugging these into the formula reveals that the rocket hits the ground at t = 0 (before launch) and t = 4 seconds. This shows how algebra predicts physical events in the real world.