Quadratic Equation Solver
Solve ax² + bx + c = 0
How the Quadratic Formula Works
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b, and c are real numbers and a is not equal to zero. The term ax² is what makes the equation quadratic: it contains a variable raised to the second power, which is why quadratic equations are also called second degree polynomial equations. They show up throughout algebra and the sciences, from calculating the path of a thrown ball to finding the dimensions of a rectangular garden with a given area.
Every quadratic equation can be solved with the quadratic formula: x equals negative b, plus or minus the square root of the discriminant, all divided by 2a. The discriminant, written as b squared minus 4ac, tells you what kind of roots to expect before you even finish the calculation. When the discriminant is positive, the equation has two distinct real roots, meaning the parabola crosses the x axis at two points. When it equals zero, there is exactly one repeated real root, and the parabola just touches the x axis at its vertex. When the discriminant is negative, there are no real roots, only a pair of complex conjugate roots, because you cannot take the square root of a negative number within the real numbers.
The plus or minus sign in the formula is what produces two answers from a single equation: one root uses addition, the other uses subtraction, and both are divided by twice the value of a. The calculator above walks through each of these steps automatically. Enter any values for a, b, and c, and it will compute the discriminant, show you the substituted formula, and display the resulting roots, whether they are two real numbers, one repeated real number, or a pair of complex numbers written in the form real part plus or minus imaginary part times i.
Quadratic equations model situations where a quantity changes at a changing rate, such as the height of a projectile over time or the profit generated by a changing price. Understanding how the discriminant predicts the number and type of solutions before solving is one of the most useful shortcuts in algebra, and it is exactly what this tool shows you at every step.