Quadratic Equation Solver
What is a Quadratic Equation?
A quadratic equation is a second-order polynomial equation in a single variable x, with the standard form ax² + bx + c = 0, where a, b, and c are coefficients and a is not equal to zero. The solutions to this equation are called the roots or zeros of the equation. A quadratic equation can have two real roots, one real root, or two complex (imaginary) roots.
How to Use the Calculator
- Identify the coefficients a, b, and c from your equation.
- Enter these values into the corresponding input fields above.
- Click the "Solve for x" button to find the roots of the equation.
The Quadratic Formula
This calculator finds the roots using the well-known quadratic formula. The formula provides the solution(s) for x.
x = (-b ± √(b² - 4ac)) / 2a
The expression inside the square root, b² - 4ac, is called the discriminant. The value of the discriminant is important because it tells us the nature of the roots before we even calculate them.
| If the Discriminant (b² - 4ac) is... | The Equation Has... |
|---|---|
| Positive (> 0) | Two distinct real roots. |
| Zero (= 0) | Exactly one real root (also called a repeated root). |
| Negative (< 0) | Two complex roots (containing an imaginary number). This calculator will indicate when this occurs by displaying the roots in the form of a + bi. For example, when you enter a=1, b=2, c=3, the discriminant is 2² - 4*1*3 = 4 - 12 = -8, which is negative. The result is x₁ = -1.0000 + 1.4142i and x₂ = -1.0000 - 1.4142i. In this case:
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